#9863402

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=hi,e^2=ghi,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, 9770598);