#472225

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16814
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339680, #72339687, #72339742
maximal normal subgroup of	#72339680, #72339687, #72339742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379421);