#8430484

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8337680);