#7857737

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

4969 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7764933);