#367329

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1110 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)(11,17)(12,24)(13,16)(14,19)(15,20)(18,26)(21,29)(22,36)(23,31)(25,30)(27,34)(28,33)(32,35)(37,40)(38,39);(5,7,8,6)(10,16,20,19)(11,21,23,28)(12,25)(17,31)(18,24,34,30)(22,37,38,35)(26,27)(29,33);(1,2,3,4)(9,11)(10,17)(12,22)(13,21)(14,28)(15,23)(16,29)(18,32)(19,33)(20,31)(24,36)(25,38)(26,35)(27,37)(30,39)(34,40);(9,12,13,26,15,25,14,27)(10,18,19,24,20,34,16,30)(11,22,21,35,23,38,28,37)(17,32,33,36,31,40,29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274525);