#116134

order	512
nilpotency class	3
isoclinism	isoclinic class of #116129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

699 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)(4,11,13,7)(5,8)(9,14,12,16)(10,15)(17,18,22,27,23,28,21,26)(19,24,32,40,30,38,31,39)(20,25,35,43,34,42,36,44)(29,45,33,48,46,37,47,41);(1,3)(2,6)(4,12,13,9)(5,10)(7,16,11,14)(8,15)(17,19)(18,24)(20,33,34,47)(21,31)(22,32)(23,30)(25,41,42,48)(26,39)(27,40)(28,38)(29,35,46,36)(37,43,45,44);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,37)(26,44)(27,43)(28,42)(30,46)(31,47)(32,33)(38,45)(39,48)(40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,f^c=fhi,g^c=gi

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, 23330);