#128376

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	#1466
prime radical	C₂⁵	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

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

normal subgroups only

This group has 562 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)(5,6,8,13,10,15,11,16)(7,17,19,12,21,26,18,28)(9,14,22,29,25,23,24,30)(20,34,33,36,35,31,32,27);(5,7,9,20,11,19,24,33,10,21,25,35,8,18,22,32)(6,12,14,27,16,26,30,34,15,28,23,36,13,17,29,31);(1,3,4,2)(5,6,8,13,10,15,11,16)(7,17,19,12,21,26,18,28)(9,23,22,30,25,14,24,29)(20,34,33,36,35,31,32,27)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,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	#72281216, #72281248, #72313774
maximal normal subgroup of	#72281216, #72281248, #72313774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35572);