#423644

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=e,e^2=i,f^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,d^b=de,e^b=ei,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330840);