#191718

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323136
maximal normal subgroup of	#72323136
maximal quotient of	#72323099

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98914);