#9945580

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

4033 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)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,22,25)(19,23,27,20)(21,29,31,24)(26,28,32,30)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(2,6)(3,9)(8,11)(10,15)(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852776);