#6843804

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#40221	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(17,18,22,25)(19,23,29,28)(20,21,24,31)(26,27,32,30);(1,2)(3,7)(4,8)(5,6)(17,19,22,29)(18,23,25,28)(20,30,24,27)(21,26,31,32);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,25,31,22,24,18,21)(19,26,23,27,29,32,28,30);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(20,24)(21,31)(26,32)(27,30);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,18,24,22,31,25,20)(19,27,28,26,29,30,23,32)

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,d^c=di,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, 6751000);