#9863958

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	#48179	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,25)(20,27)(21,24)(22,23)(26,32)(28,29)(30,31)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,34,38,41)(35,39,43,36)(37,40,47,45)(42,46,48,44);(2,3)(7,9)(8,10)(14,15)(18,23)(20,25)(24,31)(28,32)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,22,30)(18,24,23,31)(19,26,27,29)(20,28,25,32)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=ghi,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, 9771154);