#7712366

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	#39342	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5344 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,27,28,24)(21,25,31,30)(22,29,32,26);(9,11)(10,14)(12,15)(13,16)(17,19,23,28)(18,24,20,27)(21,29,31,26)(22,30,32,25);(2,5)(6,7)(10,12)(14,15)(19,28)(21,31)(24,27)(25,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,27,28,24)(21,30,31,25)(22,29,32,26);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,31)(18,25,20,30)(19,29,28,26)(22,27,32,24);(17,22,23,32)(18,26,20,29)(19,30,28,25)(21,27,31,24)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619562);