#6464278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1915 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)(17,18,21,24,22,23,30,31)(19,25,27,28,26,32,29,20);(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,20,23,28)(21,29,30,27)(24,32,31,25);(2,5)(6,8)(10,14)(11,15)(18,23)(19,26)(24,31)(27,29);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20)(18,19)(21,25)(22,28)(23,26)(24,27)(29,31)(30,32);(18,23)(20,28)(24,31)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371474);