#7672773

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579969);