#412396

order	512
nilpotency class	3
isoclinism	isoclinic class of #395552
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi

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, 319592);