#268209

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25729	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1092 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175405);