#358220

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂²×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240061
maximal normal subgroup of	#72240061
maximal quotient of	#72240039, #72240042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265416);