#9863238

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	#45833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(17,18,23,25)(19,24,28,21)(20,26,30,32)(22,31,29,27)(33,34,38,42)(35,40,41,37)(36,44,46,39)(43,45,48,47);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19,23,28)(18,24,25,21)(20,29,30,22)(26,27,32,31)(33,35)(34,37)(36,43)(38,41)(39,45)(40,42)(44,47)(46,48);(2,9)(4,11)(8,16)(12,14)(18,25)(19,28)(20,30)(27,31)(33,36,38,46)(34,39,42,44)(35,43,41,48)(37,45,40,47);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,20)(18,26)(19,29)(21,31)(22,28)(23,30)(24,27)(25,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,47)(41,48)(42,44);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,21)(18,19)(20,31)(22,32)(23,24)(25,28)(26,29)(27,30)(33,35)(34,40)(36,43)(37,42)(38,41)(39,47)(44,45)(46,48);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,22,23,29)(18,27,25,31)(19,20,28,30)(21,26,24,32)(33,37,38,40)(34,41,42,35)(36,45,46,47)(39,48,44,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=ghi,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, 9770434);