#405005

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312201);