#8184871

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	#40006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,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, 8092067);