#8188001

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	#56489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(18,20)(23,26)(24,28)(30,32)(33,34,38,37)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(19,25)(23,26)(27,31)(30,32)(35,42)(39,41)(43,48)(46,47);(6,14)(9,12)(11,13)(15,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,38,42)(34,39,37,41)(36,43,45,48)(40,46,44,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,38,45)(34,40,37,44)(35,43,42,48)(39,46,41,47);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,34)(35,39,42,41)(36,44,45,40)(43,46,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=ghi,e^2=gh,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, 8095197);