#449504

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2394 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2394 conjugacy classes of subgroups — too many to draw. Only its 218 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)(7,11,12,9)(13,14)(15,18)(16,24,25,19)(17,20)(21,27,28,26)(22,23);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,23)(14,18,20,22)(16,21,25,28)(19,26,24,27);(2,4)(13,16)(14,19)(15,21)(17,25)(18,26)(20,24)(22,27)(23,28);(7,12)(9,11)(13,14)(15,22)(16,19)(17,20)(18,23)(21,27)(24,25)(26,28)

Polycyclic presentation:

2^9|a^2=eh,b^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356700);