#7693620

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	#39491	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4747 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600816);