#8930401

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

4121 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=g,f^2=h,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, 8837597);