#6469713

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,30,21,24,20,25)(19,26,29,23,27,32,28,31);(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,31)(25,32)(26,30);(2,6)(7,8)(10,13)(14,15)(18,24)(19,27)(25,30)(28,29);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(18,24)(19,27)(25,30)(28,29);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376909);