#391830

order	512
nilpotency class	3
isoclinism	isoclinic class of #213555
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299026);