#8463862

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	#53999	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(19,20)(23,24)(27,28)(30,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=hi,f^2=ghi,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, 8371058);