#8757017

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	#54952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(19,25)(22,26)(27,31)(30,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,9)(3,12)(4,11)(10,14)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664213);