#445743

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

875 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)(5,6,11,15,10,16,9,13)(7,17,20,26,18,12,21,30)(8,22,24,14,23,28,25,29)(19,33,35,27,32,36,34,31);(5,7,9,21,10,18,11,20)(6,12,15,30,16,17,13,26)(8,19,25,34,23,32,24,35)(14,27,28,36,29,31,22,33);(6,13,16,15)(7,18)(12,26,17,30)(14,28,29,22)(19,32)(20,21)(27,36,31,33)(34,35);(2,4)(5,8,11,25,10,23,9,24)(6,14,13,28,16,29,15,22)(7,19,21,35,18,32,20,34)(12,27,30,33,17,31,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352939);