#372419

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=fgi,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, 279615);