#431491

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1087 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:

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

Polycyclic presentation:

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

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, 338687);