#194549

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1532 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 258 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101745);