#382677

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
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

1096 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289873);