#362782

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2259 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 269978);