#120510

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,e^c=eh

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, 27706);