#8095269

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	#392	C₂³
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,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, 8002465);