#8095797

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	#52916	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=hi,f^2=gh,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, 8002993);