#8460678

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	#57950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8367874);