#496741

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

576 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)(15,24)(16,27)(17,22)(18,33)(19,20)(21,31)(23,29)(25,39)(26,38)(28,42)(30,41)(32,35)(34,43)(36,40)(37,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,20,22,24)(16,28,30,25)(18,26,32,29)(21,36,37,34)(23,35,38,33)(27,39,41,42)(31,43,44,40);(6,9)(10,12)(13,16,29,43,19,28,32,44,17,30,26,40,15,25,18,31)(14,21,33,39,24,36,38,41,22,37,35,42,20,34,23,27);(5,7,8,11)(6,10,9,12)(14,22)(16,30)(20,24)(23,38)(25,28)(31,44)(33,35)(40,43)

Polycyclic presentation:

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

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, 403937);