#8696325

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	#50086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

(2,8)(4,13)(9,10)(11,15)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,2,7,8)(3,9,6,10)(4,14,13,5)(11,16,15,12)(18,23)(19,26)(25,28)(27,32)(35,44)(36,45)(40,46)(41,48);(1,3,7,6)(2,9,8,10)(4,11,13,15)(5,12,14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,4)(2,5)(3,11)(6,15)(7,13)(8,14)(9,12)(10,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,5)(2,4)(3,12)(6,16)(7,14)(8,13)(9,11)(10,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,6,7,3)(2,10,8,9)(4,11,13,15)(5,12,14,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(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=gh,c^2=gi,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, 8603521);