#8691725

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	#36839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598921);