#187611

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209967, #72209975
maximal normal subgroup of	#72209967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94807);