#346515

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62872	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2029 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253711);