#8425485

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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)(19,28)(20,30)(21,23)(22,32)(24,27)(25,31)(26,29)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,8)(3,4)(10,13)(11,12)(17,19)(18,20)(21,32)(22,25)(23,29)(24,30)(26,31)(27,28)(34,35)(39,41)(40,42)(46,47);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,20)(18,19)(21,29)(22,31)(23,32)(24,28)(25,26)(27,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,21,24,31)(18,25,27,23)(19,26,30,32)(20,29,28,22)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,22,24,29)(18,26,27,32)(19,25,30,23)(20,31,28,21);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,23,24,25)(18,21,27,31)(19,29,30,22)(20,32,28,26)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,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=dh,f^d=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, 8332681);