#7709913

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	#39735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4734 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=gh,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, 7617109);