#465836

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,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, 373032);