#370548

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 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)(9,10,14,19,15,20,13,29)(11,21,24,16,25,34,30,35)(12,26,27,18,28,17,31,36)(22,37,40,38,23,33,39,32);(1,3,5,4)(2,6,7,8)(9,11,14,24,15,25,13,30)(10,16,19,34,20,35,29,21)(12,23,27,39,28,22,31,40)(17,32,36,37,26,38,18,33);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,31)(15,28)(16,32)(18,19)(20,26)(21,37)(23,25)(24,39)(29,36)(30,40)(33,34)(35,38);(9,12)(10,18)(11,23)(13,31)(14,27)(15,28)(16,33)(17,19)(20,36)(21,38)(22,25)(24,39)(26,29)(30,40)(32,34)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,e^2=i,h^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, 277744);