#8926006

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(17,18,23,24)(19,25,20,26)(21,22,27,32)(28,31,30,29)(33,34,39,43)(35,37,41,48)(36,42,38,40)(44,47,45,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,24)(21,27)(25,26)(28,30)(33,35)(34,37)(36,44)(38,45)(39,41)(40,46)(42,47)(43,48);(2,6)(7,14)(8,11)(15,16)(17,19,23,20)(18,25,24,26)(21,28,27,30)(22,31,32,29)(33,36,39,38)(34,40,43,42)(35,44,41,45)(37,46,48,47);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20,23,19)(18,26,24,25)(21,28,27,30)(22,31,32,29);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,27)(18,22,24,32)(19,28,20,30)(25,31,26,29)(33,37)(34,41)(35,43)(36,46)(38,47)(39,48)(40,45)(42,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,27)(19,29)(20,31)(21,24)(23,32)(25,28)(26,30)(33,38,39,36)(34,42,43,40)(35,45,41,44)(37,47,48,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,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, 8833202);