#247352

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(6,11)(7,9)(8,12)(14,17)(15,18)(19,20);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,16,17)(15,18,19,20);(2,4)(5,7,10,9)(6,11,8,12)(13,15,16,19)(14,18,17,20);(5,8,10,6)(7,12,9,11)(13,14,16,17)(15,18,19,20)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279310
maximal normal subgroup of	#72279310
maximal quotient of	#72279383

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154548);