#7656572

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4503 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)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,12,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32);(1,3,5,4)(2,6,7,8)(9,10,13,12)(11,14,16,15)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32);(9,10,13,12)(11,14,16,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,2)(3,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563768);