#391613

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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

698 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)(11,15,21,32)(12,22)(13,18)(14,27)(16,25)(17,26)(19,34,23,37)(20,30,28,31)(24,38)(29,35,40,36)(33,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,21)(14,28)(16,29)(17,30)(18,32)(19,24)(20,26)(22,37)(25,36)(27,31)(33,40)(34,38)(35,39);(2,5)(6,8)(9,12)(10,16,18,33)(11,19)(13,24)(14,25)(15,29,32,40)(17,22,27,38)(20,36)(21,23)(26,39)(28,35)(30,37,31,34);(1,3,4,7)(2,6,5,8)(11,20)(12,24)(14,26)(15,30)(17,27)(19,35)(21,28)(22,38)(23,36)(29,37)(31,32)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298809);