#466639

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

905 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373835);