#410820

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1252 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)(7,16)(8,13)(9,25)(10,14)(11,15)(12,18)(17,28)(19,27)(20,31)(21,34)(22,35)(23,29)(24,30)(26,32)(33,36);(5,7,10,18,11,20,9,19)(6,12,14,31,15,27,25,16)(8,21,23,17,24,32,22,33)(13,28,29,26,30,36,35,34);(2,4)(5,8,10,23,11,24,9,22)(6,13,14,29,15,30,25,35)(7,17,19,21,20,33,18,32)(12,26,16,28,27,34,31,36);(5,6)(7,12)(8,13)(9,25)(10,14)(11,15)(16,19)(17,26)(18,31)(20,27)(21,28)(22,35)(23,29)(24,30)(32,36)(33,34)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256961, #72257167, #72257201
maximal normal subgroup of	#72256961, #72257167, #72257201

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318016);