#6435829

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(4,5)(7,8)(11,15)(14,16)(17,18,23,24,22,28,19,27)(20,30,29,21,26,31,32,25);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,28,27)(20,29,26,32)(21,30,25,31);(10,13)(14,16)(17,20,22,26)(18,25,28,21)(19,29,23,32)(24,31,27,30);(9,11,12,15)(10,14,13,16)(17,21,22,25)(18,26,28,20)(19,30,23,31)(24,32,27,29);(1,3,4,7)(2,6,5,8)(17,19,22,23)(18,27,28,24)(20,29,26,32)(21,31,25,30)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 6343025);