#353024

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260220);