#6374388

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4230 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)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(6,9)(7,12)(10,11)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(14,20)(18,21)(19,28)(26,27);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311780, #72311785, #72311795, …
maximal normal subgroup of	#72311780, #72311785, #72311795, …
maximal quotient of	#72314434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281584);