#9970096

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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)(4,5)(6,7)(9,14)(10,11)(12,13)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(2,7)(4,12)(8,14)(10,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(3,9)(8,14)(10,15)(11,16)(18,25)(19,20)(26,32)(27,28)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,3,6,9)(2,8,7,14)(4,10,12,15)(5,11,13,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,4,6,12)(2,5,7,13)(3,10,9,15)(8,11,14,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,5)(2,4)(3,11)(6,13)(7,12)(8,10)(9,16)(14,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9877292);