#7782123

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

5003 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gi,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, 7689319);