#8093319

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	#45688	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000515);