#6539614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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)(3,7)(4,8)(5,6)(10,11)(14,15)(17,18)(19,25)(20,27)(21,32)(22,29)(23,28)(24,30)(26,31);(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,25)(20,23)(21,22)(24,31)(26,30)(27,28)(29,32);(1,3,5,4)(2,7,6,8)(9,11,13,10)(12,15,16,14)(17,20,24,21)(18,26,30,25)(19,22,31,23)(27,29,32,28);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(17,21,24,20)(18,27,30,32)(19,22,31,23)(25,28,26,29);(9,12,13,16)(10,14,11,15)(17,22,24,23)(18,28,30,29)(19,20,31,21)(25,32,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6446810);