#7683413

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590609);