#10023165

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	#44172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9930361);