#422309

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329505);