#6440423

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#548	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4420 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)(5,6)(7,9)(8,10)(11,12)(14,20)(17,18)(21,25)(22,28)(23,24)(26,27);(2,4)(5,7,8,11)(6,9,10,12)(13,14)(15,25)(16,28)(17,24)(18,20)(19,23)(21,27)(22,26);(6,10)(9,12)(13,15,19,16)(14,21,23,22)(17,27,18,26)(20,25,24,28);(13,16,19,15)(14,22,23,21)(17,26,18,27)(20,28,24,25);(14,23)(20,24)(21,22)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307517, #72307781
maximal normal subgroup of	#72307517, #72307781

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347619);