#321826

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 170 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)(7,11)(8,12)(9,10)(13,14,17,27,19,23,18,22)(15,24,16,20,26,21,25,28);(1,2,3,4)(5,7,11,6,9,12,8,10)(13,15,18,25,19,26,17,16)(14,20,27,21,23,28,22,24);(2,4)(6,10)(7,12)(13,14,19,23)(15,20,26,28)(16,24,25,21)(17,22,18,27);(5,7,11,6,9,12,8,10)(13,16,18,15,19,25,17,26)(14,21,27,28,23,24,22,20)

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,d^c=dh,f^c=fi

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, 229022);