#6380827

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=dfi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288023);