#377385

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1147 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:

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

Polycyclic presentation:

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

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, 284581);