#9058567

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
prime radical	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965763);