#7673131

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	C₂×#2613	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580327);