#495558

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 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)(16,27)(17,28)(20,29)(21,30)(22,34)(23,33)(25,40)(26,39)(31,32)(35,38)(36,41)(37,44)(42,43);(2,4)(13,14,20,34,17,22,16,19,18,23,27,32,28,31,29,33)(15,21,35,44,26,37,25,30,24,36,40,43,39,42,38,41);(6,9)(10,12)(13,15,18,24)(14,21,23,36)(16,25,29,38)(17,26,28,39)(19,30,33,41)(20,35,27,40)(22,37,31,42)(32,43,34,44);(5,7,8,11)(6,10,9,12)(15,24)(21,36)(25,38)(26,39)(30,41)(35,40)(37,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,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, 402754);