#6380791

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(17,18,23,28,22,26,21,27)(19,29,20,25,30,24,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,23)(21,31)(22,30)(25,27)(26,29)(28,32);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,21)(22,31)(23,30)(24,28)(26,32)(27,29);(2,5)(6,8)(18,26)(19,30)(20,31)(27,28);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,26,28)(19,20,30,31)(24,32,29,25)

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,d^a=di,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287987);