#8425329

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	#40753	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^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, 8332525);