#7631667

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	#35761	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5408 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,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, 7538863);