#8696233

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=g,e^2=h,f^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, 8603429);