#8165193

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	#37517	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8072389);