#7806591

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4536 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,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, 7713787);