#417537

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2362 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)(5,6,11,7)(8,18,20,28)(9,13,23,16)(10,17,25,14)(12,24,15,21)(19,31,32,36)(22,30,35,27)(26,34,29,33);(5,7,11,6)(8,15,20,12)(9,16,23,13)(10,14,25,17)(18,24,28,21)(19,26,32,29)(22,27,35,30)(31,33,36,34);(5,8)(6,12)(7,15)(9,19)(10,24)(11,20)(13,26)(14,28)(16,29)(17,18)(21,25)(22,34)(23,32)(27,36)(30,31)(33,35);(2,4)(5,9)(6,13)(7,16)(8,19)(10,22)(11,23)(12,26)(14,27)(15,29)(17,30)(18,31)(20,32)(21,33)(24,34)(25,35)(28,36)

Polycyclic presentation:

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

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, 324733);