#9815139

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	#45707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=hi,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, 9722335);