#402463

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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	#72257091, #72257160, #72257164
maximal normal subgroup of	#72257091, #72257160, #72257164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309659);