#6574577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2009 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263979, #72263981, #72264061
maximal normal subgroup of	#72263979, #72263981, #72264061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481773);