#121960

order	512
nilpotency class	3
isoclinism	isoclinic class of #121960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

377 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=ei,f^c=fh,g^c=ghi

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, 29156);