#10409998

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

4290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10317194);