#181997

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1016 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=hi,g^2=i,b^a=be,d^a=di,g^a=gh,c^b=ci,e^b=eh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89193);