#331054

order	512
nilpotency class	3
isoclinism	isoclinic class of #331031
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=h,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=dei,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238250);