#386728

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
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

1077 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,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, 293924);