#473130

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 380326);