#372788

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1427 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279984);