#186735

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93931);