#137198

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=hi,d^2=gi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313762, #72313765, #72313790
maximal normal subgroup of	#72313762, #72313765, #72313790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44394);