#8720592

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	#41316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,e^2=gh,f^2=g,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, 8627788);