#8839248

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	#38510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746444);