#7682628

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

4707 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589824);