#6468948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,24,28,22,26)(19,29,30,21,25,31,32,20);(9,11)(10,13)(12,15)(14,16)(17,19,24,25)(18,20,28,21)(22,30,23,32)(26,29,27,31);(2,5)(6,8)(10,14)(11,15)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32);(11,15)(13,16)(17,21,24,20)(18,25,28,19)(22,29,23,31)(26,30,27,32);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,26,28,27)(19,30,25,32)(20,29,21,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6376144);