#493033

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×C₄×#99
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236884, #72237542, #72237546, …
maximal normal subgroup of	#72236884, #72237542, #72237546, …
maximal quotient of	#72303790, C₂×#493033, #72312339

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400229);