#6858556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28437	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=fgh,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6765752);