#381879

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1196 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=efh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=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, 289075);