#371005

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1477 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)(13,14,17,20,18,15,21,23)(16,19,24,27,25,22,26,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,20)(19,25)(21,23)(24,27)(26,28);(6,9)(10,12)(13,16,18,25)(14,19,15,22)(17,26,21,24)(20,28,23,27);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,19)(17,23)(20,21)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,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, 278201);