#476363

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1115 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1115 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 383559);