#149312

order	512
nilpotency class	4
isoclinism	isoclinic class of #129414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 944 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,e^c=ei,g^c=ghi,g^d=gi,g^e=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56508);