#6628822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(9,10,14,13)(11,15,12,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(2,6)(7,8)(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3,5,4)(2,7,6,8)(9,10,14,13)(11,16,12,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,16,13,15)(18,23)(19,26)(24,30)(25,31);(9,13,14,10)(11,15,12,16)(18,23)(19,26)(24,30)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536018);