#187652

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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	C₂×#3558	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2458 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)(5,6,9,14,10,11,8,13)(7,15,18,12,16,20,17,19)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(6,11)(8,9)(12,19)(17,18)(21,23)(22,26)(24,32)(25,31)(27,34)(28,36)(29,35)(30,33);(2,4)(5,7)(6,12)(8,18)(9,17)(10,16)(11,19)(13,20)(14,15)(24,33)(28,35)(29,36)(30,32);(7,16)(12,19)(15,20)(17,18)(22,27)(26,34)(28,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94848);