#6675654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2097 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)(15,22)(16,25)(17,20)(18,21)(19,24)(23,27)(26,28);(5,6,9,12)(7,11,8,10)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,22)(25,28);(2,4)(6,10)(7,8)(11,12);(5,7,9,8)(6,11,12,10)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(6,11)(7,8)(10,12)(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305243, #72305248, #72305278
maximal normal subgroup of	#72305243, #72305248, #72305278
maximal quotient of	#72305613, #72305678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582850);