#8695327

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	#44194	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602523);