#121956

order	512
nilpotency class	3
isoclinism	isoclinic class of #121956
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

427 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)(19,27)(20,31)(21,34)(22,26)(23,29)(24,33)(25,36)(28,44)(30,43)(32,40)(35,41)(37,45)(38,48)(39,47)(42,46);(2,7,10,9)(8,15,16,11)(17,19,21,29,22,30,36,48)(18,23,25,27,26,38,34,43)(20,28,35,47,33,46,32,45)(24,39,41,44,31,37,40,42);(1,3)(2,8,10,16)(4,12)(5,13)(6,14)(7,15,9,11)(17,20,21,35,22,33,36,32)(18,24,25,41,26,31,34,40)(19,28,29,47,30,46,48,45)(23,37,27,42,38,39,43,44)

Polycyclic presentation:

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

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, 29152);