#140894

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,11,16,12,17,10,32)(7,18,21,35,22,24,20,36)(8,23,26,28,13,33,25,14)(9,27,29,15,31,34,30,19);(5,7,9,28,10,20,29,33,12,22,31,14,11,21,30,23)(6,13,15,18,16,26,27,35,17,8,19,24,32,25,34,36);(1,2,3,4)(5,8,10,25,12,13,11,26)(6,14,16,23,17,28,32,33)(7,19,20,34,22,15,21,27)(9,24,29,36,31,18,30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48090);