#9815133

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

3979 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)(19,25)(20,27)(21,24)(22,23)(26,32)(28,29)(30,31)(33,34,39,41)(35,40,44,37)(36,42,46,48)(38,47,45,43);(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,35,39,44)(34,40,41,37)(36,45,46,38)(42,43,48,47);(18,23)(20,25)(24,31)(28,32)(34,41)(35,44)(36,46)(43,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,36)(34,42)(35,45)(37,47)(38,44)(39,46)(40,43)(41,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,22,30)(18,24,23,31)(19,26,27,29)(20,28,25,32)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,38,39,45)(34,43,41,47)(35,36,44,46)(37,42,40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 9722329);