#8097007

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	#52976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=gi,f^2=gi,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, 8004203);