#6795903

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18,23,32,21,26,22,27)(19,20,29,31,25,24,28,30);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,21,25)(18,24,26,20)(22,28,23,29)(27,31,32,30);(10,13)(14,16)(17,20,21,24)(18,25,26,19)(22,31,23,30)(27,28,32,29);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,26)(19,25)(27,32)(28,29);(1,5,6,4)(2,7,3,8)(18,26)(19,25)(27,32)(28,29)

Polycyclic presentation:

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

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, 6703099);