#8990298

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	#54925	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ghi,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,e^c=egi,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, 8897494);