#434151

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341347);