#405919

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313115);