#7183135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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)(10,11)(12,16)(17,18,20,24,21,25,22,26)(19,27,30,31,28,23,29,32);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,27)(26,31);(1,4,5,2)(3,7,8,6)(9,11,13,10)(12,15,16,14)(18,24)(19,28)(20,22)(23,31)(25,26)(27,32);(3,8)(6,7)(10,11)(14,15)(17,20,21,22)(18,24,25,26)(19,29,28,30)(23,31,27,32);(9,12,13,16)(10,14,11,15)(18,25)(19,28)(24,26)(29,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090331);