#359603

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233124, #72260670, #72260677, …
maximal normal subgroup of	#72233124, #72260670, #72260677, …
maximal quotient of	#72232827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266799);