#8924734

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

4253 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,23,27)(19,28,21,24)(20,22,26,25)(29,32,31,30)(33,34,38,39)(35,36,41,40)(37,45,47,42)(43,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32)(34,39)(36,40)(42,45)(44,48);(2,6)(4,13)(7,9)(10,16)(20,26)(22,25)(29,31)(30,32)(33,35,38,41)(34,40,39,36)(37,46,47,43)(42,44,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30)(33,35)(34,40)(36,39)(37,43)(38,41)(42,48)(44,45)(46,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,19)(18,24,27,28)(20,29,26,31)(22,32,25,30)(33,36,38,40)(34,41,39,35)(37,44,47,48)(42,46,45,43);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)(33,37)(34,42)(35,43)(36,44)(38,47)(39,45)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831930);