#180816

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
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

1118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224763, #72224769, #72227060
maximal normal subgroup of	#72224763, #72224769, #72227060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88012);