#131318

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#3199	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

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

normal subgroups only

This group has 822 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)(5,6,10,15,11,16,26,13,12,17,8,22,9,24,23,29)(7,18,19,35,20,14,30,32,21,34,27,28,25,33,36,31);(2,4)(6,13)(8,23)(9,11)(10,26)(14,28)(15,16)(17,29)(18,31)(19,30)(20,25)(22,24)(27,36)(32,34)(33,35);(5,7)(6,14)(8,19)(9,25)(10,27)(11,20)(12,21)(13,28)(15,32)(16,34)(17,33)(18,24)(22,31)(23,30)(26,36)(29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313503, #72313510, #72313521
maximal normal subgroup of	#72313503, #72313510, #72313521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38514);