#394751

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
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	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1654 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)(9,12)(13,26)(14,22)(15,23)(19,25)(21,28)(27,32)(29,33)(31,34)(35,36);(1,2)(3,4)(5,6,12,23,10,15,9,11)(7,13,24,26,16,28,18,21)(8,14,25,33,20,29,19,22)(17,27,34,35,30,36,31,32);(2,4)(5,7,9,18,10,16,12,24)(6,13,11,21,15,28,23,26)(8,17,19,31,20,30,25,34)(14,27,22,32,29,36,33,35);(5,8)(6,14)(7,17)(9,19)(10,20)(11,22)(12,25)(13,27)(15,29)(16,30)(18,31)(21,32)(23,33)(24,34)(26,35)(28,36)

Polycyclic presentation:

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

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, 301947);