#385804

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293000);