#106965

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1272 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257293, #72257298, #72257301, …
maximal normal subgroup of	#72257293, #72257298, #72257301, …
maximal quotient of	#72264364

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14161);