#8506324

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

4330 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,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, 8413520);