#358181

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265377);