#458408

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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)(6,11)(7,16)(8,20)(9,23)(12,26)(13,29)(14,15)(17,28)(18,34)(21,32)(22,35)(27,36);(5,6,9,11,10,15,23,14)(7,17,24,28,16,26,19,12)(8,13,20,25,22,29,35,31)(18,33,34,36,32,30,21,27);(5,7,9,24,10,16,23,19)(6,12,11,17,15,28,14,26)(8,21,20,18,22,34,35,32)(13,30,25,27,29,33,31,36);(2,4)(5,8,9,20,10,22,23,35)(6,13,11,25,15,29,14,31)(7,18,19,21,16,32,24,34)(12,27,26,30,28,36,17,33)

Polycyclic presentation:

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

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, 365604);