#7687577

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	#36465	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5172 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)(6,7)(9,10,13,15)(11,16,12,14)(17,18,23,25)(19,28,29,24)(20,26,32,22)(21,27,30,31);(9,11,13,12)(10,14,15,16)(17,19,23,29)(18,24,25,28)(20,30,32,21)(22,31,26,27);(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(18,25)(19,29)(20,32)(27,31);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,14,15,16)(17,20)(18,26)(19,30)(21,29)(22,25)(23,32)(24,27)(28,31);(17,21,23,30)(18,27,25,31)(19,20,29,32)(22,28,26,24);(17,22,23,26)(18,20,25,32)(19,31,29,27)(21,24,30,28)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594773);