#355558

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226668, #72226671, #72226677
maximal normal subgroup of	#72226668, #72226671, #72226677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262754);