#6966359

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,7)(4,6)(5,8)(17,18,24,29,23,28,22,27)(19,30,26,20,31,25,21,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,25)(20,27)(21,22)(23,31)(24,26)(28,30)(29,32);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,32)(18,26,28,21)(19,27,31,29)(22,25,24,30);(2,6)(5,7)(10,13)(14,16)(17,21,23,26)(18,20,28,32)(19,22,31,24)(25,27,30,29);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,29)(19,26,31,21)(20,25,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873555);