#434715

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15506
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319859, #72319962, #72320009, …
maximal normal subgroup of	#72319859, #72319962, #72320009, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341911);