#400683

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1503 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,h^c=hi,e^d=ei,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, 307879);