#456199

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1099 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)(7,9)(8,10)(11,12)(13,14)(15,23)(16,27)(17,25)(18,26)(19,22)(20,30)(21,32)(24,34)(28,40)(29,36)(31,42)(33,41)(35,44)(37,43)(38,39);(5,7,8,11)(6,9,10,12)(13,15,23,14,19,26,18,22)(16,24,34,27,30,36,29,20)(17,25,39,21,33,41,32,38)(28,40,44,42,43,37,31,35);(6,10)(9,12)(13,16,19,30)(14,20,22,27)(15,24,26,36)(17,31,33,44)(18,34,23,29)(21,37,38,40)(25,35,41,42)(28,39,43,32);(2,4)(13,17,23,39,19,33,18,32)(14,21,26,41,22,38,15,25)(16,28,34,44,30,43,29,31)(20,35,24,40,27,42,36,37)

Polycyclic presentation:

2^9|b^2=ef,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=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, 363395);