#393503

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=f,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^b=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257628, #72257651, #72257696
maximal normal subgroup of	#72257628, #72257651, #72257696
maximal quotient of	#72264353, #72264362

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300699);