#8179742

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,e^2=gh,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, 8086938);