#7815868

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

5042 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:

(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,31)(30,32);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,24)(21,29)(22,25)(26,28)(27,30)(31,32);(1,2,5,4)(3,6,8,7)(10,14)(13,15)(18,24)(19,25)(21,30)(28,31);(1,2,5,4)(3,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,30)(31,32);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,24)(21,28)(22,23)(26,29)(27,32)(30,31);(1,4,5,2)(3,7,8,6)(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,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 7723064);