#497075

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

580 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)(6,9)(7,11)(13,14)(15,25)(16,30)(17,22)(18,35)(19,36)(20,27)(21,24)(23,32)(26,40)(28,44)(29,37)(31,41)(33,34)(38,39)(42,43);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,38)(22,25)(23,26)(24,37)(30,40)(32,41)(33,44)(34,43)(35,42)(36,39);(6,9)(10,12)(13,16,32,22,19,33,34,36,17,23,30,14,21,35,18,24)(15,26,41,20,29,42,43,37,27,31,40,25,38,44,28,39);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(16,23)(24,36)(28,43)(29,38)(33,35)(40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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^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, 404271);