#9057355

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	#54005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964551);