#370546

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

1318 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)(4,8)(5,7)(9,10)(11,20)(12,24,27,16)(13,17)(14,19)(15,22)(18,29)(21,35,25,38)(23,34)(26,33,40,32)(28,31)(30,39,37,36);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,25,27,21)(13,28)(14,23)(16,30,24,37)(17,20)(18,34)(19,31)(22,29)(26,36,40,39)(32,35,33,38);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,21)(13,26)(14,27)(15,30)(17,32)(18,33)(19,24)(20,35)(22,36)(23,25)(28,39)(29,40)(31,37)(34,38);(10,17)(13,29)(15,20)(16,32)(18,19)(22,28)(24,33)(26,40)(30,35)(31,34)(36,39)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,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, 277742);