#475563

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,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,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

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, 382759);