#8115378

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8022574);