#8926158

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	#35437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833354);