#8127495

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	#59552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,40,45)(35,44,39,41)(36,38,43,47)(37,46,48,42);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(18,23)(19,26)(25,28)(27,32)(33,35,40,39)(34,41,45,44)(36,46,43,42)(37,47,48,38);(2,5)(4,14)(8,10)(11,13)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,36)(34,38)(35,46)(37,41)(39,42)(40,43)(44,48)(45,47);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,37,40,48)(34,42,45,46)(35,47,39,38)(36,44,43,41);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,38)(34,43)(35,37)(36,45)(39,48)(40,47)(41,42)(44,46);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,39,40,35)(34,44,45,41)(36,42,43,46)(37,38,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,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,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034691);