#116454

order	512
nilpotency class	3
isoclinism	isoclinic class of #98848
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

913 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 913 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

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, 23650);