#8912098

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	#59674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8819294);