#9812885

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	#41157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,37)(36,38)(39,40)(41,47)(42,46)(43,44)(45,48);(2,7)(4,8)(9,15)(12,13)(20,27)(24,30)(26,31)(28,32)(33,35)(34,37)(36,43)(38,44)(39,42)(40,46)(41,45)(47,48);(18,23)(19,25)(24,30)(26,31)(34,40)(35,42)(36,41)(44,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,41)(34,38,40,47)(35,43,42,45)(37,44,46,48);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,35)(36,44)(38,43)(39,46)(40,42)(41,48)(45,47);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38)(34,41)(35,44)(36,40)(37,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=g,f^2=g,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, 9720081);