#8926404

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=gh,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, 8833600);