#383731

order	512
nilpotency class	3
isoclinism	isoclinic class of #213818
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=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, 290927);