#7671851

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

5366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,3)(4,7,5,8)(10,14)(15,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(2,3)(7,8)(18,20)(19,25)(24,28)(27,31);(1,3,6,2)(4,8,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,4,6,5)(2,7,3,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,5,6,4)(2,8,3,7)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579047);