#496483

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

868 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)(13,14,17,23,31,39,32,41,19,21,16,22,18,24,34,43)(15,25,27,38,35,42,30,20,29,37,26,44,28,40,33,36);(5,6)(7,10)(8,9)(11,12);(2,4)(6,9)(10,12)(13,15,19,29)(14,20,21,36)(16,30,17,33)(18,35,31,28)(22,40,23,42)(24,44,39,38)(25,43,37,41)(26,34,27,32);(5,7,8,11)(6,10,9,12)(14,21)(20,36)(22,23)(24,39)(25,37)(38,44)(40,42)(41,43)

Polycyclic presentation:

2^9|a^2=g,c^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, 403679);