#7631963

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	#36218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5166 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,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, 7539159);