#8097292

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=h,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, 8004488);