#7331492

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324788
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285261
maximal normal subgroup of	#72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7238688);