#424315

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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

1508 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)(5,6,10,15,11,16,26,34)(7,17,20,28,21,33,25,12)(8,13,9,14,22,30,24,32)(18,35,36,27,19,31,23,29);(5,7)(6,12)(8,18)(9,23)(10,25)(11,21)(13,29)(14,31)(15,33)(16,28)(17,34)(19,22)(20,26)(24,36)(27,30)(32,35);(5,8,10,9,11,22,26,24)(6,13,15,14,16,30,34,32)(7,18,25,23,21,19,20,36)(12,27,33,35,28,29,17,31);(2,4)(5,9,10,22,11,24,26,8)(6,14,15,30,16,32,34,13)(7,19,20,23,21,18,25,36)(12,27,17,31,28,29,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250092, #72252980, #72256876
maximal normal subgroup of	#72250092, #72252980, #72256876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331511);