#496271

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1040 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)(5,6)(7,11)(8,10)(9,12)(15,23)(16,26)(17,28)(19,31)(21,36)(22,35)(24,41)(25,40)(27,30)(29,39)(32,43)(33,44)(34,42)(37,38);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,31)(24,33)(25,32)(26,36)(27,37)(28,35)(29,34)(30,38)(39,42)(40,43)(41,44);(7,12)(9,11)(13,15,18,25)(14,19,20,32)(16,27,29,41)(17,30,28,24)(21,37,34,44)(22,38,35,33)(23,39,40,26)(31,42,43,36);(6,10)(9,11)(14,20)(19,32)(21,34)(22,35)(31,43)(33,38)(36,42)(37,44)

Polycyclic presentation:

2^9|a^2=g,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=ce,d^b=di,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, 403467);