#8932116

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	#51954	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3891 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839312);