#6576883

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257514, #72257637, #72263869, …
maximal normal subgroup of	#72257514, #72257637, #72263869, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484079);