#121957

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,g^2=h,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, 29153);