#8425579

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	#37570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(17,18,21,22)(19,25,27,23)(20,24,29,28)(26,31,32,30)(34,39)(40,41)(42,45)(47,48);(3,10)(6,7)(11,16)(14,15)(18,22)(23,25)(24,28)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,e^2=gh,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, 8332775);