#8696019

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	#55307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=g,e^2=gh,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, 8603215);