#8095029

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	#52655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002225);