#405986

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 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)(7,16,20,17)(8,22,24,13)(9,14)(10,26)(11,15)(12,19,28,21)(18,31)(23,36,25,29)(27,32)(30,34)(33,35);(1,2)(3,4)(5,7)(6,12,15,28)(8,18)(9,19)(10,21)(11,20)(13,27,22,35)(14,16,26,17)(23,32)(24,34)(25,33)(29,30,36,31);(2,4)(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,19)(16,20)(18,31)(21,28)(22,24)(23,29)(25,36)(27,32)(30,34)(33,35);(5,8,9,23,11,24,10,25)(6,13,14,29,15,22,26,36)(7,18,19,32,20,34,21,33)(12,27,16,30,28,35,17,31)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^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, 313182);