#8752461

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

3844 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:

(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(19,25)(23,26)(27,31)(30,32)(35,41)(38,45)(42,47)(46,48);(2,6)(4,12)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(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=h,d^2=g,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, 8659657);