#8908343

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815539);