#433566

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15485
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=e,f^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,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319622, #72319935, #72320106, …
maximal normal subgroup of	#72319622, #72319935, #72320106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340762);