#495557

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,20)(17,27)(21,29)(22,37)(23,32)(25,33)(26,39)(28,34)(30,31)(35,40)(36,44)(38,43)(41,42);(2,4)(13,14,20,31,17,22,28,32,18,23,34,37,27,30,16,19)(15,21,33,42,26,36,38,40,24,35,43,44,39,41,25,29);(6,9)(10,12)(13,15,18,24)(14,21,23,35)(16,25,28,38)(17,26,27,39)(19,29,32,40)(20,33,34,43)(22,36,30,41)(31,42,37,44);(5,7,8,11)(6,10,9,12)(15,24)(21,35)(25,38)(26,39)(29,40)(33,43)(36,41)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402753);