#392122

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×2₋¹⁺⁴	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

15942 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 15942 conjugacy classes of subgroups — too many to draw. Only its 738 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) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299318);