#8696319

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	#36844	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,7)(3,12)(9,14)(10,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(35,44)(36,45)(40,46)(41,48);(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603515);