#8720425

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627621);