#420050

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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)(5,6,12,18,11,17,10,16)(7,19,9,13,22,15,21,30)(8,23,28,33,27,14,26,34)(20,32,25,36,35,31,24,29);(5,7)(6,13)(8,24)(9,10)(11,22)(12,21)(14,31)(15,16)(17,30)(18,19)(20,26)(23,32)(25,27)(28,35)(29,33)(34,36);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,29)(15,32)(16,33)(17,23)(18,34)(19,31)(21,24)(22,35)(30,36);(5,9,12,22,11,21,10,7)(6,15,18,30,17,19,16,13)(8,25,28,35,27,24,26,20)(14,32,34,36,23,31,33,29)

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,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, 327246);