#426741

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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)(5,6,10,14,11,15,26,17)(7,9,13,23,18,24,30,35)(8,19,22,29,21,31,25,12)(16,32,34,27,33,36,28,20);(5,7,10,13,11,18,26,30)(6,9,14,23,15,24,17,35)(8,16,25,28,21,33,22,34)(12,27,31,32,29,20,19,36);(5,8,10,25,11,21,26,22)(6,12,14,31,15,29,17,19)(7,16,13,28,18,33,30,34)(9,20,23,36,24,27,35,32);(2,4)(5,9)(6,13)(7,17)(8,20)(10,23)(11,24)(12,28)(14,18)(15,30)(16,19)(21,27)(22,32)(25,36)(26,35)(29,34)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250604, #72252967, #72256873
maximal normal subgroup of	#72250604, #72252967, #72256873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333937);