#9863951

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	#44171	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9771147);