#8750253

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
prime radical	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,e^2=gh,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, 8657449);