#7683405

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
prime radical	#59056	C₂
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) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590601);