#457019

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 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)(7,10)(8,9)(11,12)(13,14,18,22,17,21,31,38)(15,23,26,40,25,19,33,35)(16,27,30,37,29,39,32,20)(24,41,42,34,28,44,43,36);(6,9)(10,12)(13,15,18,26,17,25,31,33)(14,19,22,35,21,23,38,40)(16,28,30,43,29,24,32,42)(20,36,27,41,37,34,39,44);(7,11)(10,12)(13,16,17,29)(14,20,21,37)(15,24,25,28)(18,32,31,30)(19,34,23,36)(22,39,38,27)(26,43,33,42)(35,41,40,44);(2,4)(5,7,8,11)(6,10,9,12)(16,29)(20,37)(24,28)(27,39)(30,32)(34,36)(41,44)(42,43)

Polycyclic presentation:

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

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, 364215);