#460414

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 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)(3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,27)(16,22)(17,23)(18,25)(19,24)(20,26)(21,30)(28,40)(29,41)(31,39)(32,38)(33,37)(34,42)(35,43)(36,44);(5,7,9,12)(6,10,8,11)(13,15,18,30,20,32,19,31)(14,21,24,27,26,39,25,38)(16,33,34,28,35,17,29,36)(22,40,41,37,43,44,42,23);(2,4)(6,8)(10,11)(13,16)(14,22)(15,28)(17,31)(18,29)(19,34)(20,35)(21,37)(23,39)(24,42)(25,41)(26,43)(27,40)(30,33)(32,36)(38,44);(5,8,9,6)(7,11,12,10)(13,17,18,36,20,33,19,28)(14,23,24,40,26,37,25,44)(15,29,30,16,32,34,31,35)(21,22,27,41,39,43,38,42)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 367610);