#8096635

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003831);