#129472

order	512
nilpotency class	4
isoclinism	isoclinic class of #129405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1004 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 106 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)(5,6,9,13,10,14,8,12)(7,11,15,18,16,19,17,20)(21,22)(23,27)(24,31,32,26)(25,28)(29,35,36,33)(30,34);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,18)(14,19)(22,26,28,31)(24,32)(27,33,34,35)(29,36);(2,4)(5,8,10,9)(6,12,14,13)(7,15,16,17)(11,18,19,20)(21,23,25,30)(22,27,28,34)(24,29,32,36)(26,33,31,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,f^b=fg,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, 36668);