#8695881

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=gh,e^2=g,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, 8603077);