#9910056

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

4020 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817252);