#7735685

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

4831 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7642881);