#115281

order	512
nilpotency class	3
isoclinism	isoclinic class of #115274
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,22,26,23,28,21,27)(19,29,31,24,32,40,38,41)(20,33,35,25,37,43,36,44)(30,46,34,42,48,39,47,45);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,24)(20,34)(21,38)(22,31)(23,32)(25,42)(26,40)(27,29)(28,41)(30,36)(33,46)(35,48)(37,47)(39,43)(44,45);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,37)(18,25,28,44)(19,30,32,48)(21,35,22,36)(24,39,41,46)(26,33,27,43)(29,45,40,42)(31,47,38,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,d^c=di,e^c=eh,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, 22477);