#364220

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271416);