#497024

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 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)(14,20)(15,24)(16,19)(18,26)(21,34)(23,39)(25,37)(27,42)(28,41)(30,40)(31,32)(33,35)(36,44)(38,43);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14,17,20)(15,25,28,21)(16,29,19,22)(18,23,32,33)(24,34,41,37)(26,35,31,39)(27,36,30,38)(40,44,42,43);(7,12)(9,11)(13,15,26,40,19,27,32,41,17,28,31,42,16,30,18,24)(14,21,35,43,22,38,33,34,20,25,39,44,29,36,23,37);(6,10)(9,11)(13,16,17,19)(14,22,20,29)(15,27,28,30)(18,31,32,26)(21,36,25,38)(23,35,33,39)(24,40,41,42)(34,43,37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fh,d^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=ce,d^b=di,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, 404220);