#8425575

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(18,20)(19,25)(24,28)(27,29)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(3,10)(6,7)(11,16)(14,15)(17,18,22,20)(19,26,25,23)(21,24,30,28)(27,32,29,31)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19,22,25)(18,23,20,26)(21,29,30,27)(24,32,28,31)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,24,20,28)(19,27,25,29)(23,31,26,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332771);