#7842489

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

4838 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)(10,13)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,30)(31,32);(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,26,27)(20,25,30,32)(21,24,31,29);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(19,28)(21,31)(23,27)(25,32);(1,3)(2,4)(5,7)(6,8)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,28)(31,32);(1,4)(2,3)(5,8)(6,7)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,29)(28,32);(17,21,22,31)(18,25,26,32)(19,29,28,24)(20,27,30,23)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,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, 7749685);