#8115367

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

3781 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)(3,11)(4,15)(5,7)(6,16)(8,14)(9,13)(10,12)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,14)(2,8,5,11)(4,6,13,12)(9,16,15,10)(20,28)(23,30)(26,32)(29,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2,7,5)(3,8,14,11)(4,15,13,9)(6,10,12,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,2)(3,8,14,11)(4,15,13,9)(6,16,12,10)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022563);