#7623180

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60649	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530376);