#6461249

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,30,21,24,20,25)(19,26,28,23,27,31,29,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(18,24)(19,27)(25,30)(28,29);(11,15)(13,16)(18,24)(19,27)(25,30)(28,29);(17,20,21,22)(18,25,24,30)(19,28,27,29)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=fg,c^2=gh,e^2=i,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, 6368445);