#411163

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265308, #72307302, #72307361
maximal normal subgroup of	#72265308, #72307302, #72307361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318359);