#7633277

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4754 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)(10,15)(11,16)(17,18,22,21)(19,23,27,26)(20,24,30,29)(25,31,28,32);(9,10,14,15)(11,12,16,13)(18,21)(23,26)(24,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,22,21)(19,23,27,26)(20,24,30,29)(25,31,28,32);(9,12,14,13)(10,16,15,11)(17,19,22,27)(18,23,21,26)(20,28,30,25)(24,32,29,31);(9,12,14,13)(10,16,15,11)(17,20,22,30)(18,24,21,29)(19,25,27,28)(23,31,26,32);(1,3,4,7)(2,5,6,8)(9,13,14,12)(10,11,15,16)(17,21,22,18)(19,26,27,23)(20,29,30,24)(25,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=hi,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, 7540473);