#137729

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 112 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)(5,6,8,11,9,7,10,12)(13,14)(15,23,25,26)(16,20)(17,27,28,21)(18,22)(19,24);(5,7,8,12,9,6,10,11)(13,15,17,24,18,25,28,20)(14,19,27,23,22,16,21,26);(2,4)(13,16,18,19)(14,20,22,24)(15,21,25,27)(17,26,28,23)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fg,g^b=gi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44925);