#469692

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1082 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:

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

Polycyclic presentation:

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

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, 376888);