#138359

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,f^d=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, 45555);