#498271

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

536 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)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,23,25,42)(16,27,30,41)(18,32,29,22)(19,34,35,24)(21,38,37,31)(26,43,33,39)(28,40,44,36);(1,3,4,2)(6,9)(7,11);(13,15,17,25)(14,19,20,35)(16,28,30,44)(18,33,29,26)(21,39,37,43)(22,40,32,36)(23,41,42,27)(24,38,34,31);(5,7,8,11)(6,10,9,12)(14,20)(19,35)(21,37)(22,32)(23,42)(27,41)(36,40)(39,43)

Polycyclic presentation:

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

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, 405467);