#495526

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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

844 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402722);