#370547

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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)(4,8)(5,7)(9,10)(11,21)(12,24,27,17)(13,28)(14,30)(15,20)(16,23)(18,29)(19,31)(22,35,33,38)(25,36,32,39)(26,34,40,37);(1,3,5,4)(2,6,7,8)(9,11,15,18)(10,16,20,13)(12,22)(14,28,31,23)(17,25)(19,29,30,21)(24,32)(26,39)(27,33)(34,35)(36,40)(37,38);(1,4,5,3)(2,6,7,8)(9,12,15,27)(10,17,20,24)(11,22,18,33)(13,25,16,32)(14,26,31,40)(19,34,30,37)(21,35,29,38)(23,36,28,39);(9,13,15,16)(10,18,20,11)(12,25,27,32)(14,21,31,29)(17,33,24,22)(19,23,30,28)(26,35,40,38)(34,36,37,39)

Polycyclic presentation:

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

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, 277743);