#395188

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1788 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)(8,20)(9,12)(14,29)(15,25)(17,32)(18,19)(21,22)(23,27)(26,35)(34,36);(1,2)(3,4)(5,6,12,25,10,15,9,11)(7,13,16,27,19,28,18,23)(8,14,21,30,22,29,20,24)(17,26,31,35,32,36,33,34);(2,4)(5,7)(6,13)(8,17)(9,18)(10,19)(11,23)(12,16)(14,26)(15,28)(20,33)(21,31)(22,32)(24,34)(25,27)(29,36)(30,35);(5,8,9,20,10,22,12,21)(6,14,11,24,15,29,25,30)(7,17,18,33,19,32,16,31)(13,26,23,34,28,36,27,35)

Polycyclic presentation:

2^9|b^2=ei,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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, 302384);