#8445262

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(11,16)(13,14)(17,18,22,26)(19,23,24,20)(21,25,30,29)(27,31,32,28)(33,34)(35,41)(36,47)(37,43)(38,48)(39,44)(40,45)(42,46);(1,2,6,10)(3,5,9,7)(4,8,12,15)(11,13,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35,40,38)(34,41,45,48)(36,39,46,43)(37,47,44,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,36)(34,42)(35,39)(37,41)(38,43)(40,46)(44,48)(45,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(33,37,40,44)(34,43,45,39)(35,42,38,47)(36,41,46,48);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,38,40,35)(34,41,45,48)(36,43,46,39)(37,47,44,42);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,39,40,43)(34,44,45,37)(35,46,38,36)(41,42,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352458);