#478245

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=fi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^d=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, 385441);