#365774

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^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.

Relationships

maximal subgroup of	#72226668, #72226675, #72226677
maximal normal subgroup of	#72226668, #72226675, #72226677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272970);