#9814955

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	#52587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,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, 9722151);