#421527

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

955 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=e,e^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,e^b=ei,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328723);