#7846448

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

4984 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7753644);