#134455

order	512
nilpotency class	4
isoclinism	isoclinic class of #124657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 518 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) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=gi,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=eh,f^c=fghi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41651);