#10340229

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	#40244	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(18,24)(19,20)(22,26)(28,30)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(2,3)(5,13)(7,9)(12,16)(17,18,23,24)(19,27,20,25)(21,26,32,22)(28,29,30,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(4,11)(7,9)(12,16)(14,15)(17,19,23,20)(18,25,24,27)(21,28,32,30)(22,29,26,31)(33,35,38,43)(34,39,41,36)(37,42,46,45)(40,47,48,44);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,23,19)(18,25,24,27)(21,28,32,30)(22,31,26,29)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,26)(19,28)(20,30)(22,24)(23,32)(25,31)(27,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,26)(18,21,24,32)(19,29,20,31)(25,28,27,30)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10247425);