#180817

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂×C₄³
socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,d^a=dh,c^b=ci,d^c=dh

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224763, #72224784, #72285505, …
maximal normal subgroup of	#72224763, #72224784, #72285505, …
maximal quotient of	#72321669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88013);