#7839991

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

4877 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=g,f^2=i,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, 7747187);