#7905876

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7813072);