#8926628

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

3971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,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, 8833824);