#8107591

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014787);