#355552

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di

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, 262748);