#6909937

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308665, #72308849
maximal normal subgroup of	#72308665, #72308849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817133);