#400790

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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250020, #72257156, #72257218
maximal normal subgroup of	#72250020, #72257156, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307986);