#184047

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286208, #72286355, #72322005, …
maximal normal subgroup of	#72286208, #72286355, #72322005, …
maximal quotient of	#72287262, #72319616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91243);