#365442

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1472 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)(11,22)(12,17)(13,30)(14,20)(15,21)(16,25)(18,29)(19,32)(23,36)(24,33)(26,34)(27,40)(28,37)(31,35)(38,39);(5,7,8,6)(9,11)(10,16)(12,27)(13,24)(14,31)(15,26)(17,36)(18,33)(19,34)(20,22)(21,35)(23,29)(25,32)(28,39)(30,38)(37,40);(1,2,3,4)(9,12,15,13)(10,17,21,30)(11,23,26,39)(14,28,32,29)(16,33,35,40)(18,20,37,19)(22,36,34,38)(24,31,27,25);(9,13,15,12)(10,18,21,37)(11,24,26,27)(14,28,32,29)(16,33,35,40)(17,20,30,19)(22,38,34,36)(23,31,39,25)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,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,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, 272638);