#8137134

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	#40292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044330);