#8107293

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	#43373	C₂
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014489);