#353006

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
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×#1721	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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)(10,15)(12,14)(16,19)(18,20)(21,22)(23,31)(24,27)(25,35)(26,29)(28,33)(30,32)(34,36);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(21,23,32,34,26,33,25,24)(22,27,35,28,29,36,30,31);(5,6)(7,10)(8,12)(9,11)(13,15)(14,17)(16,18)(19,20)(21,24,25,33,26,34,32,23)(22,28,30,27,29,31,35,36);(1,2,3,4)(6,11)(10,18)(12,17)(13,19)(22,29)(23,33)(24,34)(30,35)

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=di,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, 260202);