#499661

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,9,16,32,15,34,17,12,18,31,46,11,19,10,33)(7,20,23,38,35,39,59,40,22,49,58,41,25,47,24,13)(8,26,27,45,36,50,63,43,30,48,62,14,29,44,28,51)(21,53,55,65,60,42,67,66,54,64,68,52,57,37,56,61);(1,2)(3,4)(5,7,12,22)(6,13,18,40)(8,21,30,54)(9,24,31,59)(10,23,34,58)(11,35,32,25)(14,42,45,37)(15,41,19,38)(16,47,46,39)(17,49,33,20)(26,61,48,66)(27,56,62,67)(28,55,63,68)(29,60,36,57)(43,64,51,53)(44,65,50,52);(2,4)(5,8,12,30)(6,14,18,45)(7,21,22,54)(9,28,31,63)(10,27,34,62)(11,36,32,29)(13,37,40,42)(15,43,19,51)(16,48,46,26)(17,50,33,44)(20,52,49,65)(23,56,58,67)(24,55,59,68)(25,60,35,57)(38,64,41,53)(39,66,47,61);(6,15,18,19)(7,22)(9,31)(10,34)(13,38,40,41)(14,43,45,51)(16,33,46,17)(20,47,49,39)(21,54)(25,35)(26,50,48,44)(27,62)(28,63)(37,64,42,53)(52,61,65,66)(57,60)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ef,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406857);