#8930314

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=hi,e^2=h,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, 8837510);