#6669434

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2125 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305265, #72305276, #72305280
maximal normal subgroup of	#72305265, #72305276, #72305280
maximal quotient of	#72305942, #72305985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576630);