#388356

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,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, 295552);