#9815148

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=hi,e^2=g,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, 9722344);