#8291629

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	#45536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(3,10,5,8)(4,13,15,6)(11,16,14,12)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(34,40)(36,45)(41,42)(43,46);(1,3,7,5)(2,8,9,10)(4,14,15,11)(6,12,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,41)(36,38)(37,42)(39,40)(43,47)(44,46)(45,48);(2,9)(6,13)(8,10)(12,16)(19,25)(23,26)(27,31)(30,32)(33,35,39,37)(34,41,40,42)(36,43,45,46)(38,47,48,44);(1,4)(2,6)(3,11)(5,14)(7,15)(8,16)(9,13)(10,12)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36)(34,38)(35,43)(37,46)(39,45)(40,48)(41,47)(42,44);(1,5,7,3)(2,10,9,8)(4,11,15,14)(6,16,13,12)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,35)(34,42,40,41)(36,43,45,46)(38,47,48,44);(1,6)(2,4)(3,12)(5,16)(7,13)(8,14)(9,15)(10,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8198825);