#7672739

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	#35634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5258 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)(3,7,4,8)(9,10,13,15)(11,12,16,14)(18,23)(21,29)(24,30)(25,31);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,18,22,23)(19,24,27,30)(20,25,28,31)(21,26,29,32);(1,4,6,3)(2,7,5,8)(9,12,13,14)(10,11,15,16)(18,23)(20,28)(24,30)(26,32);(1,5,6,2)(3,8,4,7)(9,11)(10,12)(13,16)(14,15)(17,19,22,27)(18,24,23,30)(20,26,28,32)(21,25,29,31);(17,20,22,28)(18,25,23,31)(19,26,27,32)(21,24,29,30);(17,21,22,29)(18,26,23,32)(19,25,27,31)(20,24,28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579935);