#372418

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1217 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,20)(12,26)(13,18,21,31)(15,29)(16,28)(17,30)(22,37,23,33)(24,34)(25,36)(27,35)(32,39,38,40);(9,11,21,24,14,25,13,29)(10,15,31,20,19,34,18,36)(12,23,27,39,28,22,30,40)(16,32,17,33,26,38,35,37);(2,5)(6,8)(9,12)(10,16)(11,22)(13,27)(14,28)(15,32)(17,18)(19,26)(20,37)(21,30)(23,25)(24,39)(29,40)(31,35)(33,36)(34,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,23)(13,30)(14,28)(15,33)(16,18)(19,35)(20,38)(21,27)(22,25)(24,39)(26,31)(29,40)(32,36)(34,37)

Polycyclic presentation:

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

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, 279614);