#7710492

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	#45451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,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, 7617688);