#6380766

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

1883 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^c=ei,e^d=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, 6287962);