#497782

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ehi,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, 404978);