#354477

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,g^2=i,b^a=be,c^a=cf,f^a=fi,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321530, #72342922, #72342923, …
maximal normal subgroup of	#72321530, #72342922, #72342923, …
maximal quotient of	#72343036, #72343064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261673);