#129337

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293047, #72293128, #72293133
maximal normal subgroup of	#72293047, #72293128, #72293133

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36533);