#6873880

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 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)(17,18,22,27,23,28,21,26)(19,24,29,32,25,20,30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,25)(18,24,28,20)(21,30,22,29)(26,31,27,32);(3,8)(5,7)(10,14)(15,16)(17,20,23,24)(18,25,28,19)(21,31,22,32)(26,30,27,29);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,22)(18,26,28,27)(19,29,25,30)(20,31,24,32);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,28,26)(19,30,25,29)(20,31,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fhi,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, 6781076);