#131319

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38515);