#476990

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384186);