#6437229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2944 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 2944 conjugacy classes of subgroups — too many to draw. Only its 626 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) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250494, #72250602, #72257032
maximal normal subgroup of	#72250494, #72250602, #72257032
maximal quotient of	#72264688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344425);