#9815398

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(3,8)(4,5)(6,15)(7,11)(9,14)(10,12)(13,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34,39,44)(35,45,47,40)(36,41,37,42)(38,43,48,46);(1,3,7,13)(2,8,11,16)(4,12,14,6)(5,10,9,15)(19,25)(23,26)(27,29)(31,32)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(1,4,7,14)(2,9,11,5)(3,6,13,12)(8,10,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36,39,37)(34,41,44,42)(35,46,47,43)(38,40,48,45);(1,4)(2,5)(3,12)(6,13)(7,14)(8,10)(9,11)(15,16)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,5)(2,4)(3,10)(6,16)(7,9)(8,12)(11,14)(13,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37,39,36)(34,42,44,41)(35,46,47,43)(38,40,48,45);(1,6,7,12)(2,10,11,15)(3,4,13,14)(5,8,9,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,48)(34,43,44,46)(35,42,47,41)(36,40,37,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=gh,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, 9722594);