#6549492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#37839	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456688);