#498614

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efhi,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,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, 405810);