#7174292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fg,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308866, #72308949
maximal normal subgroup of	#72308866, #72308949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7081488);