#6667478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283468, #72283897, #72284109, …
maximal normal subgroup of	#72283468, #72283897, #72284109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574674);