#496371

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^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,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, 403567);