#232446

order	512
nilpotency class	3
isoclinism	isoclinic class of #232445
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2175 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:

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

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, 139642);