#496485

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

870 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)(7,11)(13,14,16,22,28,36,40,34,18,21,29,35,17,23,27,20)(15,19,24,32,38,43,44,41,26,31,39,42,25,33,37,30);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,21,31)(16,24,29,39)(17,25,28,38)(20,30,34,41)(22,32,35,42)(23,33,36,43)(27,37,40,44);(2,4)(6,9)(10,12)(14,20)(16,27)(17,28)(19,30)(21,34)(22,23)(24,37)(25,38)(29,40)(31,41)(32,33)(35,36)(39,44)(42,43);(5,7,8,11)(6,10,9,12)(14,21)(19,31)(20,34)(22,35)(23,36)(30,41)(32,42)(33,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ce,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403681);