#348693

order	512
nilpotency class	3
isoclinism	isoclinic class of #348674
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3460	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

889 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)(3,7)(4,8)(5,6)(9,15)(10,12)(11,13)(14,16)(17,18,23,28)(19,29,31,40)(20,34,36,42)(21,27,22,26)(24,33,39,32)(25,37,41,35)(30,44,45,38)(43,47,48,46);(2,6)(7,12)(8,13)(14,15)(17,19,22,33,23,31,21,32)(18,24,27,40,28,39,26,29)(20,30,37,47,36,45,35,46)(25,38,42,48,41,44,34,43);(1,3)(2,7)(4,9)(5,10)(6,12)(8,14)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,42)(28,41)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(1,4,5,11)(2,8,6,13)(3,9,10,16)(7,14,12,15)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

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, 255889);