#351859

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 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)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20)(21,22)(23,32)(24,34)(25,29)(26,31)(27,28)(30,33)(35,36);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19)(21,23,26,27)(22,28,31,32)(24,35,25,33)(29,36,34,30);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(21,24,26,25)(22,29,31,34)(23,33,27,35)(28,30,32,36);(21,25,26,24)(22,30,31,36)(23,33,27,35)(28,29,32,34)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259055);