#7777974

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

4812 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)(9,10)(11,16)(12,14)(13,15)(17,18)(19,27)(20,25)(21,30)(22,24)(23,28)(26,31)(29,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,21)(22,28)(24,27)(25,26)(29,31)(30,32);(1,3,5,4)(2,6,8,7)(10,15)(14,16)(18,24)(19,28)(20,29)(26,30);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,20,22,29)(18,25,24,32)(19,21,28,31)(23,26,27,30);(1,3,5,4)(2,6,8,7)(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,20)(22,31)(23,25)(24,30)(27,32)(28,29);(1,4,5,3)(2,7,8,6)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,e^2=gi,f^2=gh,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, 7685170);