#9815096

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,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, 9722292);