#8928720

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

4142 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)(3,10,5,8)(6,12,14,9)(11,15,13,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,4)(6,14)(8,10)(11,13)(19,27)(20,28)(22,30)(24,32)(33,34,38,37)(35,39,43,41)(36,44,45,40)(42,47,48,46);(1,4,7,2)(3,10,5,8)(6,12,14,9)(11,16,13,15)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,37,38,34)(35,39,43,41)(36,40,45,44)(42,47,48,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^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, 8835916);