#381136

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
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

1117 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288332);