#455093

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1579 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)(7,16)(8,22,24,36)(9,26)(10,14)(11,15)(12,19)(13,25,29,23)(17,21)(18,32,34,27)(20,28)(30,35,31,33);(1,2)(3,4)(5,7,9,19,11,21,10,20)(6,12,14,16,15,28,26,17)(8,18,23,33,24,34,25,35)(13,27,22,30,29,32,36,31);(2,4)(5,6)(7,17)(8,22)(9,26)(10,14)(11,15)(12,20)(13,23)(16,21)(18,27)(19,28)(24,36)(25,29)(30,35)(31,33)(32,34);(5,8,9,23,11,24,10,25)(6,13,14,22,15,29,26,36)(7,18,19,33,21,34,20,35)(12,27,16,30,28,32,17,31)

Polycyclic presentation:

2^9|a^2=f,b^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,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,f^d=fi

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, 362289);