#497916

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	C₂²×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 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)(6,9)(10,12)(13,14)(15,19)(16,27)(17,31)(18,23)(20,30)(21,25)(22,26)(24,38)(28,41)(29,42)(32,34)(33,39)(35,43)(36,44)(37,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,23,22)(16,28,29,24)(17,25,32,40)(20,35,36,33)(21,34,37,31)(27,41,42,38)(30,43,44,39);(5,7,8,11)(6,10,9,12)(13,16,25,39,15,24,32,44,18,29,40,43,26,28,17,30)(14,20,31,41,22,35,37,42,23,36,34,38,19,33,21,27);(6,9)(10,12)(16,29)(20,36)(24,28)(27,42)(30,44)(33,35)(38,41)(39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=efh,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 405112);