#421876

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1185 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:

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

Polycyclic presentation:

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

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, 329072);