#356549

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 conjugacy classes of subgroups; only the 458 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 458 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(10,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(9,11,13,12)(10,15,14,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,3,4,2)(9,12,13,11)(10,15,14,16)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ei,d^2=egi,f^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263745);