#9850770

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	#50034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9757966);