#369093

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1944 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:

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

Polycyclic presentation:

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

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, 276289);