#373656

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2161 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)(11,21)(12,26)(13,20)(14,19)(15,24)(16,28)(17,27)(18,31)(22,37)(23,32)(25,34)(29,36)(30,33)(35,39)(38,40);(9,11,14,25)(10,15,19,33)(12,23,28,38)(13,30,31,24)(16,35,26,37)(17,32,36,40)(18,34,20,21)(22,27,39,29);(2,4)(5,7,8,6)(9,12)(10,16)(11,22)(13,27)(14,28)(15,32)(17,18)(19,26)(20,36)(21,35)(23,30)(24,38)(25,39)(29,31)(33,40)(34,37);(9,12)(10,17)(11,23)(13,29)(14,28)(15,32)(16,18)(19,36)(20,26)(21,37)(22,30)(24,39)(25,38)(27,31)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280852);