#9040934

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(19,27)(25,30)(26,29)(33,34,40,38)(35,44,43,39)(36,42,37,41)(45,48,47,46);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,18,22,23)(19,28,27,24)(20,30,31,25)(21,26,32,29)(33,35)(34,39)(36,45)(37,47)(38,44)(40,43)(41,48)(42,46);(2,6)(4,13)(7,9)(10,14)(20,31)(21,32)(25,30)(26,29)(33,36,40,37)(34,41,38,42)(35,45,43,47)(39,48,44,46);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,19,22,27)(18,24,23,28)(20,26,31,29)(21,25,32,30)(33,37,40,36)(34,42,38,41)(35,45,43,47)(39,48,44,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,20,22,31)(18,25,23,30)(19,29,27,26)(21,28,32,24)(33,38,40,34)(35,44,43,39)(36,41,37,42)(45,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,21)(18,26)(19,25)(20,28)(22,32)(23,29)(24,31)(27,30)(33,39,40,44)(34,43,38,35)(36,46,37,48)(41,45,42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948130);