#358979

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 418 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,d^2=gi,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=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, 266175);