#412866

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320062);