#136348

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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) = 512

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

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

Polycyclic presentation:

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

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, 43544);