#357374

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=egi,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,h^a=hi,c^b=cef,d^b=de,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 264570);