#429928

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 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:

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

Polycyclic presentation:

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

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, 337124);