#441079

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1459 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348275);