#131324

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#3197	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38520);