#434714

order	512
nilpotency class	3
isoclinism	isoclinic class of #419463
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 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)(3,13,16,8)(4,17,20,28)(5,11)(6,12)(7,10)(9,19,26,22)(14,25)(15,27,21,24)(18,29)(23,30)(31,32);(1,3)(2,8)(4,18)(5,15)(6,21)(7,16)(9,25)(10,13)(11,24)(12,27)(14,19)(17,23)(20,30)(22,31)(26,32)(28,29);(1,4,5,19,7,20,6,22)(2,9,12,28,10,26,11,17)(3,14,15,30,16,31,21,18)(8,23,27,25,13,29,24,32);(8,13)(23,29)(24,27)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237064, #72237066, #72237661, …
maximal normal subgroup of	#72237064, #72237066, #72237661, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341910);