#141171

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,f^d=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, 48367);