#180815

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₄³
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1174 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:

(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(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);(15,21)(18,24)(22,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224763, #72224771, #72224772, …
maximal normal subgroup of	#72224763, #72224771, #72224772, …
maximal quotient of	#72227016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88011);