#395981

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1325 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)(7,19)(8,22)(9,25,27,31)(10,17)(11,16)(12,18)(13,21)(14,24)(15,26,32,28)(20,29,35,36)(23,30,33,34);(1,2)(3,4)(5,7,10,21,12,22,11,14)(6,13,16,19,18,24,17,8)(9,20,26,34,27,35,28,30)(15,29,31,33,32,36,25,23);(2,4)(5,8,11,24,12,19,10,13)(6,14,17,22,18,21,16,7)(9,23,28,36,27,33,26,29)(15,30,25,35,32,34,31,20);(5,9)(6,15)(7,20)(8,23)(10,26)(11,28)(12,27)(13,29)(14,30)(16,31)(17,25)(18,32)(19,33)(21,34)(22,35)(24,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303177);