#137158

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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×#2010	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

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

normal subgroups only

This group has 770 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,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,f^c=fh,e^d=ei,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, 44354);