#8093215

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

4130 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,45)(43,47)(46,48);(2,7)(4,8)(9,13)(12,15)(19,25)(20,27)(23,28)(24,29)(34,39)(35,42)(36,45)(46,47);(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,42)(34,40,39,37)(36,43,45,48)(41,47,44,46);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,38,40)(34,42,39,35)(36,46,45,47)(41,48,44,43)

Polycyclic presentation:

2^9|c^2=i,e^2=gh,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, 8000411);