#449745

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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)(6,9)(10,12)(13,14,17,22,18,23,32,20)(15,19,24,34,27,36,26,35)(16,28,29,38,31,37,30,21)(25,40,41,43,42,44,39,33);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,25)(17,24)(18,27)(20,35)(21,33)(22,34)(23,36)(26,32)(28,40)(29,41)(30,39)(31,42)(37,44)(38,43);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,24)(17,32)(21,28)(22,23)(25,39)(26,27)(29,30)(34,35)(37,38)(40,44)(41,42);(6,9)(10,12)(13,16,18,31)(14,21,23,38)(15,25,27,42)(17,30,32,29)(19,33,36,43)(20,28,22,37)(24,39,26,41)(34,44,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 356941);