#183473

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,d^b=di,e^b=eh,f^b=fh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210308, #72210369, #72210396, …
maximal normal subgroup of	#72210308, #72210369, #72210396, …
maximal quotient of	#72210369, #72320205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90669);