#8188019

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	#42712	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095215);