#417676

order	512
nilpotency class	3
isoclinism	isoclinic class of #417674
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2409 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238736, #72256170, #72256186, …
maximal normal subgroup of	#72238736, #72256170, #72256186, …
maximal quotient of	#72238738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324872);