#7806674

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

5021 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713870);