#121944

order	512
nilpotency class	3
isoclinism	isoclinic class of #121940
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

419 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,22,28,23,26,21,27)(19,29,32,44,31,24,37,39)(20,25,36,43,34,41,35,42)(30,38,33,40,46,45,47,48);(2,7,10,9)(8,15,16,11)(17,19)(18,24,26,29)(20,33)(21,37)(22,32)(23,31)(25,40,41,48)(27,44,28,39)(30,35)(34,47)(36,46)(38,43,45,42);(1,3)(2,8,10,16)(4,12)(5,13)(6,14)(7,15,9,11)(17,20)(18,25,26,41)(19,30)(21,36)(22,35)(23,34)(24,38,29,45)(27,43,28,42)(31,46)(32,47)(33,37)(39,48,44,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29140);