#9815000

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

4125 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=hi,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, 9722196);