#8902510

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	#57189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8809706);