#366309

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1902 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)(3,4)(5,6,8,7)(9,10)(11,16)(12,24)(13,29)(14,18)(15,19)(17,26)(20,33)(21,31)(22,38)(23,32)(25,39)(27,34)(28,35)(30,37)(36,40);(2,4)(5,7,8,6)(9,11,14,23,15,21,13,22)(10,16,18,32,19,31,29,38)(12,25,27,36,28,37,26,20)(17,33,24,39,34,40,35,30);(9,12,15,28)(10,17,19,34)(11,20,21,36)(13,27,14,26)(16,30,31,39)(18,35,29,24)(22,25,23,37)(32,40,38,33);(2,4)(11,21)(16,31)(20,36)(22,23)(25,37)(30,39)(32,38)(33,40)

Polycyclic presentation:

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

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, 273505);