#493012

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1275 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)(5,6,10,14)(7,15,18,29)(8,12,19,26)(9,21,22,13)(11,23,25,17)(16,28,30,36)(20,33,32,27)(24,34,35,31);(5,7)(6,11)(8,16)(9,17,22,23)(10,18)(12,24)(13,15,21,29)(14,25)(19,30)(20,31,32,34)(26,35)(27,28,33,36);(1,3,4,2)(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(28,36)(31,34);(5,8,10,19)(6,12,14,26)(7,16,18,30)(9,20,22,32)(11,24,25,35)(13,27,21,33)(15,28,29,36)(17,31,23,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=i,b^a=be,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400208);