#497780

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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)(7,9)(8,10)(11,12)(13,14,16,21,28,36,40,32,18,23,29,31,17,22,26,19)(15,20,24,34,38,44,41,42,25,33,37,43,27,35,39,30);(5,7,8,11)(6,9,10,12)(14,19)(15,24)(16,26)(17,28)(21,22)(23,32)(25,37)(27,41)(29,40)(30,34)(31,36)(35,44)(38,39)(42,43);(6,10)(9,12)(15,25)(20,33)(24,37)(27,38)(30,42)(34,43)(35,44)(39,41);(13,15,26,39,17,27,29,37,18,25,40,41,28,38,16,24)(14,20,19,30,22,35,31,43,23,33,32,42,36,44,21,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404976);