#128765

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35961);