#9794368

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	#55601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=gh,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, 9701564);