#401036

order	512
nilpotency class	3
isoclinism	isoclinic class of #394108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14811
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1752 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)(3,14,17,23)(4,9)(5,11)(6,13)(7,12)(8,16,10,18)(15,29,31,28)(19,25)(20,26)(21,27)(22,32,24,30);(1,3,5,18,7,17,6,16)(2,8,11,23,12,10,13,14)(4,15,19,32,20,31,21,30)(9,24,25,28,26,22,27,29);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,22)(10,24)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(18,32)(23,29);(1,3,5,18,7,17,6,16)(2,10,11,14,12,8,13,23)(4,15,19,32,20,31,21,30)(9,24,25,28,26,22,27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237073, #72237076, #72237661, …
maximal normal subgroup of	#72237073, #72237076, #72237661, …
maximal quotient of	#72251210, C₂×#401036, #72262489

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308232);