#8621613

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	#38510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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)(4,5)(6,11)(7,9)(10,15)(12,16)(13,14)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,40,43)(35,44,46,41)(36,39,37,47)(38,42,48,45);(1,3,7,12)(2,8,9,16)(4,13,10,11)(5,14,15,6)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,40,46)(34,41,43,44)(36,42,37,45)(38,47,48,39);(2,9)(3,12)(5,15)(11,13)(19,24)(21,29)(27,30)(28,32)(33,36,40,37)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,4,7,10)(2,5,9,15)(3,11,12,13)(6,16,14,8)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,37,40,36)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,5)(2,10)(3,6)(4,9)(7,15)(8,13)(11,16)(12,14)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,38)(34,42)(35,39)(36,41)(37,44)(40,48)(43,45)(46,47);(1,6)(2,11)(3,5)(4,8)(7,14)(9,13)(10,16)(12,15)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,39)(34,37)(35,38)(36,43)(40,47)(41,45)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528809);