#470602

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

869 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)(5,6,10,16,11,15,26,34)(7,12,21,30,19,31,20,29)(8,22,24,33,9,14,25,13)(17,35,23,27,36,32,18,28);(5,7,11,19)(6,12)(8,23,9,18)(10,20,26,21)(13,32)(14,28)(15,31)(16,29)(17,25,36,24)(22,27)(30,34)(33,35);(2,4)(5,8,11,9)(6,13,15,33)(7,17,19,36)(10,25,26,24)(12,27,31,28)(14,34,22,16)(18,20,23,21)(29,32,30,35);(5,9,11,8)(6,14,15,22)(7,18,19,23)(10,24,26,25)(12,28,31,27)(13,16,33,34)(17,21,36,20)(29,35,30,32)

Polycyclic presentation:

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

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, 377798);