#449588

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2036 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)(7,15,18,11)(8,12)(9,21)(10,13)(14,22)(16,25,23,29)(17,31,34,26)(19,30)(20,27)(24,28)(32,35,33,36);(7,16)(9,22)(11,25)(14,21)(15,29)(17,33)(18,23)(19,24)(26,36)(28,30)(31,35)(32,34);(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,15)(14,29)(16,22)(19,32)(20,34)(21,25)(24,33)(27,31)(28,35)(30,36);(2,4)(5,8,10,20)(6,12,13,27)(7,17,18,34)(9,24,22,19)(11,26,15,31)(14,30,21,28)(16,32,23,33)(25,35,29,36)

Polycyclic presentation:

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

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, 356784);