#7144052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051248);