#493084

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#442
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,14,11,17,26,16)(7,18,22,30,23,34,20,36)(8,13,10,27,25,15,28,35)(12,29,32,21,33,24,31,19);(2,4)(5,7,10,21,9,20,25,29,11,23,28,19,26,22,8,24)(6,12,15,18,16,32,27,30,17,33,13,34,14,31,35,36);(5,8,9,10,11,25,26,28)(6,13,14,27,17,15,16,35)(7,19,22,29,23,21,20,24)(12,30,32,34,33,36,31,18);(5,9,11,26)(6,14,17,16)(7,20,23,22)(8,10,25,28)(12,31,33,32)(13,27,15,35)(18,36,34,30)(19,24,21,29)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225127, #72239482, #72239532, …
maximal normal subgroup of	#72225127, #72239482, #72239532, …
maximal quotient of	#72225095, #72225106, #72265628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400280);