#416372

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1008 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)(7,12,20,31)(8,22,19,13)(9,26)(10,16)(11,28)(14,23)(15,17)(18,29)(21,35,24,32)(25,33,34,30)(27,36);(1,2,4,3)(5,7,17,24,10,20,11,21)(6,12,15,32,16,31,28,35)(8,23,34,9,19,18,25,27)(13,29,33,36,22,14,30,26);(1,3,4,2)(5,8,10,19)(6,13)(7,18,20,23)(9,24,27,21)(11,25,17,34)(12,29)(14,31)(15,33)(16,22)(26,35)(28,30)(32,36);(5,9,11,23,10,27,17,18)(6,14,15,26,16,29,28,36)(7,19,21,34,20,8,24,25)(12,30,32,13,31,33,35,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72226807, #72226810
maximal normal subgroup of	#72224916, #72226807, #72226810
maximal quotient of	#72224908

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323568);