#7758303

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

4960 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:

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

Polycyclic presentation:

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

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, 7665499);