#8445359

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	#42893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352555);