#120858

order	512
nilpotency class	3
isoclinism	isoclinic class of #120858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
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

387 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)(3,8,4,14)(5,12)(6,10)(7,11)(9,15,16,13)(17,18)(19,23,28,38)(20,30)(21,34)(22,25)(24,32)(26,35)(27,41,31,46)(29,45,44,37)(33,47)(36,48,39,43)(40,42);(1,3,6,13,5,4,7,15)(2,8,10,9,12,14,11,16)(17,19,21,29,22,28,35,44)(18,23,26,37,25,38,34,45)(20,31,32,48,33,27,42,43)(24,39,30,46,40,36,47,41);(1,4,5,3)(2,9)(6,15,7,13)(8,11)(10,14)(12,16)(17,20)(18,24,25,40)(19,27,28,31)(21,32)(22,33)(23,36)(26,30,34,47)(29,43,44,48)(35,42)(37,41)(38,39)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fhi,g^b=gh,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, 28054);