#360131

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(5,6)(7,8)(9,10,15,19,13,18,14,29)(11,20,23,32,22,16,30,33)(12,24,28,36,26,17,27,35)(21,37,25,34,38,31,39,40);(1,2,4,3)(9,11,13,22)(10,16,18,20)(12,25,26,39)(14,30,15,23)(17,34,24,40)(19,33,29,32)(21,28,38,27)(31,36,37,35);(6,8)(9,12,13,26)(10,17,18,24)(11,21,22,38)(14,28,15,27)(16,31,20,37)(19,36,29,35)(23,39,30,25)(32,40,33,34);(1,3,4,2)(10,18)(16,20)(17,24)(19,29)(31,37)(32,33)(34,40)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267327);