#113335

order	512
nilpotency class	3
isoclinism	isoclinic class of #113330
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20531);