#7812014

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

4831 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=g,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, 7719210);