#417666

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60371	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2323 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324862);