#367067

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1438 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)(5,6,7,8)(9,10,14,20)(11,21,24,17)(12,25,26,31)(13,16,15,19)(18,28,34,27)(22,33,29,30)(23,37,39,40)(32,36,35,38);(1,2)(3,4)(10,16,20,19)(11,22,24,29)(12,26)(17,21)(18,25,34,31)(23,38,39,36)(27,28)(30,33);(2,4)(6,8)(9,11)(10,17)(12,23)(13,22)(14,24)(15,29)(16,30)(18,32)(19,33)(20,21)(25,40)(26,39)(27,36)(28,38)(31,37)(34,35);(9,12,13,27,14,26,15,28)(10,18,19,25,20,34,16,31)(11,23,22,36,24,39,29,38)(17,32,33,40,21,35,30,37)

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,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, 274263);