#10195830

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	#53837	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103026);