#7061383

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#23573	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,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=eh,d^c=dhi,e^c=ei,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282284
maximal normal subgroup of	#72282284

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6968579);