#423671

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1558 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 330867);