#8093312

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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)(3,8,5,4)(6,9,14,16)(11,15,13,12)(17,18,23,26)(19,22,29,24)(20,25,30,21)(27,28,31,32)(36,43)(40,45)(42,47)(46,48);(1,3,7,5)(2,4,10,8)(6,13,14,11)(9,15,16,12)(20,30)(21,25)(27,31)(28,32)(34,38)(35,41)(40,45)(42,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,19,23,29)(18,24,26,22)(20,27,30,31)(21,28,25,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5,7,3)(2,8,10,4)(6,13,14,11)(9,15,16,12)(17,20)(18,25)(19,27)(21,26)(22,28)(23,30)(24,32)(29,31);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,21,23,25)(18,20,26,30)(19,28,29,32)(22,31,24,27)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,22)(18,19)(20,28)(21,31)(23,24)(25,27)(26,29)(30,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=ghi,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, 8000508);