#7631670

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
prime radical	#36259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5080 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538866);