#6514493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	#35521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 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)(15,24)(16,27)(17,22)(18,28)(19,23)(20,26)(21,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,16,17,25)(14,20,22,24,23,21,28,27);(2,4)(6,9)(7,11)(13,16,18,25,19,15,17,26)(14,21,22,27,23,20,28,24);(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,21)(17,28)(18,22)(19,23)(24,26)(25,27);(6,9)(10,12)(13,17,19,18)(14,22,23,28)(15,25,16,26)(20,24,21,27)

Polycyclic presentation:

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

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, 6421689);