#7002199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,e^2=f,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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257437, #72257553, #72257641
maximal normal subgroup of	#72257437, #72257553, #72257641
maximal quotient of	#72264679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6909395);