#6433419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1941 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,26)(21,32)(22,29)(23,28)(24,27)(25,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,22)(23,31)(26,29)(27,32)(28,30);(11,15)(13,16)(17,20,23,21)(18,25,28,30)(19,22,31,24)(26,29,32,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,20)(18,26,28,32)(19,22,31,24)(25,29,30,27);(17,22,23,24)(18,27,28,29)(19,20,31,21)(25,26,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340615);