#7806589

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

4656 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:

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

Polycyclic presentation:

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

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, 7713785);