#368844

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1700 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)(5,6,8,7)(9,10)(11,16)(12,24)(13,15)(14,19)(17,25)(18,21)(20,28)(22,36)(23,29)(26,32)(27,33)(30,37)(31,35)(34,39)(38,40);(5,7,8,6)(10,15)(11,20)(13,21)(16,29)(17,24)(18,19)(22,34)(23,35)(25,26)(30,39)(32,33)(37,38);(1,2,3,4)(9,11,21,35,14,23,13,20)(10,16,18,31,19,29,15,28)(12,22,25,37,27,38,26,34)(17,30,33,40,32,39,24,36);(9,12,14,27)(10,17,19,32)(11,22,23,38)(13,26,21,25)(15,24,18,33)(16,30,29,39)(20,34,35,37)(28,36,31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgi,d^2=i,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,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, 276040);