#496819

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
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	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

748 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cei,f^c=fh,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, 404015);