#6606430

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513626);