#7631137

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

5238 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538333);