#7035355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^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=dhi,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313656, #72313674, #72314037
maximal normal subgroup of	#72313656, #72313674, #72314037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6942551);