#395187

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
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1740 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)(7,15)(8,20)(9,23)(13,29)(14,26)(17,33)(18,19)(21,22)(24,28)(27,36)(32,35);(1,2)(3,4)(5,6,9,11,10,14,23,26)(7,16,18,28,19,12,15,24)(8,13,20,25,22,29,21,30)(17,32,34,36,33,27,31,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,24)(13,27)(14,16)(15,23)(20,34)(21,31)(22,33)(25,35)(26,28)(29,32)(30,36);(5,8,9,20,10,22,23,21)(6,13,11,25,14,29,26,30)(7,17,18,34,19,33,15,31)(12,27,24,35,16,32,28,36)

Polycyclic presentation:

2^9|b^2=e,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, 302383);