#8097235

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

4086 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004431);