#7307926

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
prime radical	#24248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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	#72285038, #72285261
maximal normal subgroup of	#72285038, #72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215122);