#8434133

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	#52375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,f^2=g,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, 8341329);