#358975

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,d^2=g,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg

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, 266171);