#8107668

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,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, 8014864);