#464963

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1133 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 372159);