#390591

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

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

814 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)(5,6,11,16,12,9,10,15)(7,17,21,13,20,29,19,26)(8,22,24,31,25,14,27,30)(18,33,34,36,23,32,35,28);(5,7)(6,13)(8,23,25,18)(9,26)(10,21)(11,19)(12,20)(14,28,22,36)(15,29)(16,17)(24,34,27,35)(30,32,31,33);(1,3,4,2)(5,8,11,24,12,25,10,27)(6,14,15,31,9,22,16,30)(7,18,19,35,20,23,21,34)(13,28,29,33,26,36,17,32);(5,9,12,6)(7,17,20,29)(8,22,25,14)(10,15,11,16)(13,21,26,19)(18,32,23,33)(24,30,27,31)(28,34,36,35)

Polycyclic presentation:

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

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, 297787);