#6843788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42839	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,28,29,24)(20,25,30,21)(23,27,31,32);(1,2)(3,7)(4,8)(5,6)(17,19,22,29)(18,24,26,28)(20,31,30,23)(21,32,25,27);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,23,19,22,30,31,29)(18,25,27,28,26,21,32,24);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,26)(21,25)(24,28)(27,32);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,31,28,22,25,23,24)(18,20,32,29,26,30,27,19)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6750984);