#374061

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2494 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,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, 281257);