#6454482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#24366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2151 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)(7,10)(8,9)(11,12)(15,21)(18,27)(22,28)(23,26);(2,4)(13,14,16,20)(15,22,21,28)(17,25,24,19)(18,26,27,23);(6,9)(10,12)(13,15)(14,18)(16,21)(17,26)(19,28)(20,27)(22,25)(23,24);(14,19)(17,24)(18,28)(20,25)(22,27)(23,26);(5,7,8,11)(6,10,9,12)(15,21)(18,27)(22,28)(23,26)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256930, #72256973, #72256997, …
maximal normal subgroup of	#72256930, #72256973, #72256997, …
maximal quotient of	#72264164, #72264661, #72264780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361678);