#498616

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=efh,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, 405812);