#128402

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=i,f^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ch,d^b=dg,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313487, #72313559
maximal normal subgroup of	#72313487, #72313559

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35598);