#209182

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×#1655	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1218 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(6,10)(7,8)(11,12)(14,17)(15,19)(18,20);(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,18)(17,19);(5,6)(7,10)(8,12)(9,11)(15,19)(17,20);(1,3,4,2)(6,11)(10,12)(14,18)(17,20)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72217670, #72217679, #72217689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116378);