#8754967

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3763 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662163);