#116202

order	512
nilpotency class	3
isoclinism	isoclinic class of #116201
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	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

580 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)(3,9,12,6)(4,13)(5,7)(8,15)(10,14,16,11)(17,18,21,26,22,25,33,39)(19,23,29,35,28,36,41,46)(20,24,32,38,31,37,44,47)(27,34,40,45,30,42,43,48);(1,2)(3,10)(4,13)(5,7)(6,11)(8,15)(9,14)(12,16)(17,19)(18,23)(20,30,31,27)(21,29)(22,28)(24,34,37,42)(25,36)(26,35)(32,43,44,40)(33,41)(38,45,47,48)(39,46);(1,3)(2,6)(4,14)(5,12)(7,9)(8,16)(10,13)(11,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,37)(26,38)(28,30)(29,40)(33,44)(35,45)(36,42)(39,47)(41,43)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23398);