#8729057

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,26,21,24)(20,25,31,30)(22,27,32,29)(35,43)(36,45)(39,46)(40,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,28,26)(20,27,31,29)(22,30,32,25)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(3,10)(6,7)(11,16)(14,15)(17,19,23,21)(18,24,28,26)(20,27,31,29)(22,30,32,25)(33,35,38,43)(34,39,42,46)(36,44,45,37)(40,47,48,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,31)(18,25,28,30)(19,29,21,27)(22,24,32,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,26,28,24)(20,27,31,29)(22,30,32,25)(33,36)(34,40)(35,44)(37,43)(38,45)(39,47)(41,46)(42,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22)(18,27)(19,30)(20,26)(21,25)(23,32)(24,31)(28,29)(33,37,38,44)(34,41,42,47)(35,36,43,45)(39,40,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8636253);