#8197114

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,14)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34)(35,41)(36,47)(37,43)(38,48)(39,44)(40,45)(42,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(18,24)(19,20)(26,29)(27,28)(33,35,40,38)(34,41,45,48)(36,39,46,43)(37,47,44,42);(2,7)(6,9)(8,12)(14,15)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36)(34,42)(35,39)(37,41)(38,43)(40,46)(44,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37,40,44)(34,43,45,39)(35,42,38,47)(36,41,46,48);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,38,40,35)(34,41,45,48)(36,43,46,39)(37,47,44,42);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,39,40,43)(34,44,45,37)(35,46,38,36)(41,42,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=ghi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8104310);