#7384258

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1512 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=h,d^2=i,e^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=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308774, #72308985, #72309041
maximal normal subgroup of	#72308774, #72308985, #72309041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7291454);