#8197117

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104313);