#477211

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

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

normal subgroups only

This group has 845 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=e,d^2=ei,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, 384407);