#476213

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

842 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 383409);