#123891

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214149, #72215281, #72215990, …
maximal normal subgroup of	#72214149, #72215281, #72215990, …
maximal quotient of	#72214123, #72215276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31087);