#8810317

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

4072 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(2,7)(5,14)(8,11)(9,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(3,12)(4,13)(8,11)(9,15)(17,18,23,22)(19,24,21,26)(20,28,30,25)(27,32,29,31)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3,6,12)(2,8,7,11)(4,10,13,16)(5,9,14,15)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4,6,13)(2,9,7,15)(3,10,12,16)(5,8,14,11)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,5,6,14)(2,10,7,16)(3,9,12,15)(4,8,13,11)(17,22,23,18)(19,24,21,26)(20,28,30,25)(27,31,29,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=g,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, 8717513);