#421900

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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)(13,14,17,22,18,23,19,24)(15,25,33,36,28,20,27,39)(16,29,34,21,32,40,31,38)(26,41,42,37,30,44,43,35);(5,7)(6,10)(8,11)(9,12)(13,15,19,27,18,28,17,33)(14,20,24,36,23,25,22,39)(16,30,31,42,32,26,34,43)(21,37,29,41,38,35,40,44);(5,8,9,6)(7,11,12,10)(13,16)(14,21)(15,26)(17,31)(18,32)(19,34)(20,35)(22,29)(23,38)(24,40)(25,37)(27,42)(28,30)(33,43)(36,41)(39,44);(2,4)(7,12)(10,11)(13,17,18,19)(14,22,23,24)(15,27,28,33)(16,31,32,34)(20,36,25,39)(21,29,38,40)(26,42,30,43)(35,41,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329096);