#7782121

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7689317);