#382602

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1357 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1357 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289798);