#382673

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

Polycyclic presentation:

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

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, 289869);