#6576168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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)(5,6,8,9)(7,10,11,12)(13,14,20,16)(15,21,25,27)(17,22,28,26)(18,23,19,24);(6,9)(10,12)(13,15,20,25)(14,21,16,27)(17,24,28,23)(18,26,19,22);(2,4)(13,16,20,14)(15,17,25,28)(18,23,19,24)(21,22,27,26);(5,7,8,11)(6,10,9,12)(13,17,20,28)(14,22,16,26)(15,24,25,23)(18,27,19,21);(13,18)(14,23)(15,25)(16,24)(19,20)(21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256952, #72256969, #72257004
maximal normal subgroup of	#72256952, #72256969, #72257004
maximal quotient of	#72264865, #72264970, #72265186, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483364);