#367376

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274572);