#497137

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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)(7,10)(8,11)(9,12)(15,23)(16,24)(17,27)(19,29)(21,30)(22,34)(25,35)(26,39)(28,38)(31,44)(32,42)(33,43)(36,37)(40,41);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,14,18,20)(15,19,25,32)(16,21,28,33)(17,22,27,34)(23,29,35,42)(24,30,38,43)(26,31,36,40)(37,41,39,44);(5,8,10,9)(6,11,7,12)(13,15,24,37,17,26,28,35,18,25,38,39,27,36,16,23)(14,19,30,41,22,31,33,42,20,32,43,44,34,40,21,29);(5,9,10,8)(6,11,7,12)(14,20)(15,25)(21,33)(22,34)(23,35)(26,36)(30,43)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efh,d^2=e,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, 404333);