#399173

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

1348 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)(7,13)(8,21,23,26)(9,14,25,24)(10,15)(11,27)(12,16)(17,34,30,18)(19,31)(20,33)(22,35,36,29)(28,32);(1,2)(3,4)(5,7,10,19,12,20,11,28)(6,13,15,31,16,33,27,32)(8,22,24,17,23,36,14,30)(9,18,26,35,25,34,21,29);(2,4)(5,8,10,24,12,23,11,14)(6,9,15,26,16,25,27,21)(7,17,19,36,20,30,28,22)(13,29,31,18,33,35,32,34);(5,9)(6,14)(7,18)(8,15)(10,26)(11,21)(12,25)(13,30)(16,24)(17,33)(19,35)(20,34)(22,31)(23,27)(28,29)(32,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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	#72256961, #72257164, #72257218
maximal normal subgroup of	#72256961, #72257164, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306369);