#8751961

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

4179 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:

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

Polycyclic presentation:

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

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, 8659157);