#8127211

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	#36758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,39)(35,36,43,40)(37,45,46,41)(42,47,48,44);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(34,39)(36,40)(41,45)(44,47);(2,8)(3,9)(4,13)(11,16)(19,26)(20,28)(22,25)(23,30)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20)(18,23)(19,27)(21,28)(22,29)(24,30)(25,31)(26,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^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, 8034407);