#8734913

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(34,35)(39,41)(40,42)(46,47);(3,11)(4,12)(7,16)(8,13)(18,25)(20,24)(26,31)(27,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642109);