#181755

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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22)(23,31)(24,28)(25,29)(26,30)(27,34)(32,36)(33,35);(5,7)(6,11,14,20)(8,17)(9,16)(10,18)(12,15,13,19)(21,23)(22,27,30,31)(24,32)(25,33)(26,34)(28,35,29,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,31,36);(5,8,10,9)(6,13,14,12)(7,17,18,16)(11,19,20,15)(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=hi,d^2=hi,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, 88951);