#122199

order	512
nilpotency class	3
isoclinism	isoclinic class of #122199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,23,28,37)(20,30)(21,33)(22,25)(24,32)(26,34)(27,40,42,45)(29,44,43,36)(31,46)(35,48,38,41)(39,47);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,23,25,37)(20,27)(21,29)(22,28)(24,38,39,35)(26,36,33,44)(30,45,46,40)(31,42)(32,41)(34,43)(47,48);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20)(18,24,25,39)(19,27)(21,32)(22,31)(23,35,37,38)(26,30,33,46)(28,42)(29,41)(34,47)(36,40,44,45)(43,48)

Polycyclic presentation:

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

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, 29395);