#6383753

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2428	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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)(11,14)(12,16)(13,15)(17,18,22,26,21,25,20,24)(19,27,30,31,29,23,28,32);(1,2,5,6)(3,7,4,8)(9,11,14,10)(12,16,13,15)(17,18,20,24,21,25,22,26)(19,23,28,31,29,27,30,32);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,19,21,29)(18,23,25,27)(20,28,22,30)(24,31,26,32);(1,3,5,4)(2,8,6,7)(9,13,14,12)(10,15,11,16)(17,20,21,22)(18,24,25,26)(19,28,29,30)(23,31,27,32);(1,4,5,3)(2,8,6,7)(19,29)(23,27)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^b=dfi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290949);