#497941

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

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

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, 405137);