#9815025

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=ghi,e^2=g,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, 9722221);