#8734094

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	#38543	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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=g,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, 8641290);