#7631946

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	#35821	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5217 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,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, 7539142);