#370982

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 278178);