#9863073

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(34,40)(35,37)(42,46)(43,45);(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,5,6,10)(2,8,9,3)(4,13,14,16)(7,15,12,11)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(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=h,d^2=i,e^2=ghi,f^2=h,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, 9770269);