#8425349

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	#37493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,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, 8332545);