#10040775

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947971);