#415597

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250020, #72257164, #72257214
maximal normal subgroup of	#72250020, #72257164, #72257214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322793);