#7722497

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

4970 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)(19,25)(20,21)(22,24)(23,27)(26,30)(28,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,28)(21,31)(22,27)(24,25)(26,29)(30,32);(2,5)(6,7)(18,24)(19,27)(20,30)(29,31);(9,11,13,12)(10,15,14,16)(17,19)(18,25)(20,28)(21,29)(22,27)(23,24)(26,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,28)(21,24)(22,30)(23,29)(25,31)(27,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,21,22,26)(18,20,24,30)(19,29,27,31)(23,32,25,28)

Polycyclic presentation:

2^9|d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7629693);