#7782711

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	#34741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4620 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18)(19,24)(20,26)(21,30)(22,27)(23,25)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,27)(21,29)(22,26)(23,28)(25,31)(30,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(19,28)(20,30)(22,29);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,15,14,16)(17,20,23,30)(18,26,25,21)(19,27,28,32)(22,31,29,24);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,21,23,26)(18,20,25,30)(19,29,28,22)(24,27,31,32);(17,22,23,29)(18,27,25,32)(19,26,28,21)(20,31,30,24)

Polycyclic presentation:

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

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, 7689907);