#9935833

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 9843029);