#8926701

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

4151 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833897);