#512178

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2206	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

681 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419374);