#473514

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,15)(7,9)(8,20,24,13)(10,14,26,17)(12,29)(16,25)(18,27)(19,28)(21,35)(22,30)(23,33,36,31)(32,34);(1,2,3,4)(5,7)(6,12)(8,21)(9,17)(10,29)(11,19)(13,22)(14,28)(15,27)(16,24)(18,26)(20,34)(23,32)(25,33)(30,36)(31,35);(2,4)(5,8,10,23,11,24,26,36)(6,13,14,31,15,20,17,33)(7,16,18,32,19,21,29,30)(9,25,27,34,28,35,12,22);(5,9,26,27,11,28,10,12)(6,7,17,18,15,19,14,29)(8,22,36,25,24,34,23,35)(13,30,33,16,20,32,31,21)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380710);