#394808

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=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, 302004);