#400510

order	512
nilpotency class	3
isoclinism	isoclinic class of #400510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14802
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319853
maximal normal subgroup of	#72319853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307706);