#8184901

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	#57849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 8092097);