#8818140

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725336);