#470600

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 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)(5,6,10,14,9,15,23,12)(7,16,19,26,20,11,17,27)(8,13,22,28,21,29,24,30)(18,31,32,35,34,25,33,36);(5,7,9,20)(6,11)(8,18,21,34)(10,17,23,19)(12,27)(13,25)(14,26)(15,16)(22,33,24,32)(28,35)(29,31)(30,36);(2,4)(6,12)(7,17)(8,21)(10,23)(13,28)(14,15)(18,32)(19,20)(25,31)(26,27)(29,30)(33,34);(5,8,9,21)(6,13,15,29)(7,18,20,34)(10,24,23,22)(11,25,16,31)(12,28,14,30)(17,32,19,33)(26,36,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 377796);