#8202262

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,9,11,4)(5,8,13,14)(10,15,16,12)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(2,7)(4,9)(8,14)(12,15)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(5,13)(8,14)(10,16)(12,15)(18,24)(21,32)(23,27)(25,29)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,5,6,13)(2,8,7,14)(3,10,11,16)(4,12,9,15)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

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

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, 8109458);