#181750

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1664 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)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20)(21,22)(23,31)(24,28)(25,29)(26,30)(27,34)(32,36)(33,35);(5,7)(6,10,11,18)(8,15)(9,14)(12,13,16,19)(17,20)(21,23)(22,27,30,31)(24,32)(25,33)(26,34)(28,35,29,36);(7,14)(10,18)(13,19)(15,20)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,31,36);(6,11)(10,18)(12,16)(13,19)(21,25,26,24)(22,29,30,28)(23,33,34,32)(27,36,31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88946);