#496487

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

854 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)(13,14,19,24,27,20,28,39,17,22,18,23,15,25,29,38)(16,30,34,37,26,44,35,21,32,42,33,40,31,43,36,41);(5,6)(7,10)(8,9)(11,12)(13,15,17,27)(14,20,22,25)(16,31,32,26)(18,28,19,29)(21,40,41,37)(23,38,24,39)(30,44,42,43)(33,35,34,36);(2,4)(6,9)(10,12)(13,16,17,32)(14,21,22,41)(15,26,27,31)(18,35,19,36)(20,37,25,40)(23,43,24,44)(28,33,29,34)(30,38,42,39);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=ei,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, 403683);