#8750025

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

4128 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)(6,9,15,16)(10,13,12,14)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34)(35,43)(36,42)(37,41)(38,44)(39,40)(45,48)(46,47);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(34,40)(36,46)(41,43)(44,45);(2,8)(3,4)(9,16)(10,12)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,37)(34,41,40,43)(36,44,46,45)(38,42,48,47);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,46)(34,42,40,47)(35,45,37,44)(38,43,48,41);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37,39,35)(34,43,40,41)(36,44,46,45)(38,42,48,47);(1,6,7,15)(2,9,8,16)(3,10,4,12)(5,13,11,14)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,41,46,43)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657221);